Patent Publication Number: US-11379301-B2

Title: Fractional redundant array of silicon independent elements

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     Priority benefit claims for this application are made in the accompanying Application Data Sheet, Request, or Transmittal (as appropriate, if any). To the extent permitted by the type of the instant application, this application incorporates by reference for all purposes the following applications, all commonly owned with the instant application at the time the invention was made:
         U.S. Provisional Application (Ser. No. 61/418,846), filed Dec. 1, 2010, first named inventor Jeremy Isaac Nathaniel WERNER, and entitled DYNAMIC HIGHER-LEVEL REDUNDANCY MODE MANAGEMENT WITH INDEPENDENT SILICON ELEMENTS;   U.S. Provisional Application (Ser. No. 61/433,918), filed Jan. 18, 2011, first named inventor Jeremy Isaac Nathaniel WERNER, and entitled HIGHER-LEVEL REDUNDANCY INFORMATION COMPUTATION;   PCT Application (Serial No. PCTIUS11/062726), filed Nov. 30, 2011 first named inventor Jeremy Isaac Nathaniel WERNER, and entitled DYNAMIC HIGHER-LEVEL REDUNDANCY MODE MANAGEMENT WITH INDEPENDENT SILICON ELEMENTS;   PCT Application (Serial No. PCT/US12/21682), filed Jan. 18, 2012, first named inventor Jeremy Isaac Nathaniel WERNER, and entitled HIGHER-LEVEL REDUNDANCY INFORMATION COMPUTATION; and   U.S. Provisional Application (Ser. No. 61/682,561), filed Aug. 13, 2012, first named inventor Earl T. COHEN, and entitled FRACTIONAL REDUNDANT ARRAY OF SILICON INDEPENDENT ELEMENTS.       

    
    
     BACKGROUND 
     Field 
     Advancements in storage technology and manufacturing are needed to provide improvements in cost, profitability, performance, efficiency, and utility of use. 
     Related Art 
     Unless expressly identified as being publicly or well known, mention herein of techniques and concepts, including for context, definitions, or comparison purposes, should not be construed as an admission that such techniques and concepts are previously publicly known or otherwise part of the prior art. All references cited herein (if any), including patents, patent applications, and publications, are hereby incorporated by reference in their entireties, whether specifically incorporated or not, for all purposes. 
     Synopsis 
     The invention may be implemented in numerous ways, e.g., as a process, an article of manufacture, an apparatus, a system, a composition of matter, and a computer readable medium such as a computer readable storage medium (e.g., media in an optical and/or magnetic mass storage device such as a disk, an integrated circuit having non-volatile storage such as flash storage), or a computer network wherein program instructions are sent over optical or electronic communication links. The Detailed Description provides an exposition of one or more embodiments of the invention that enable improvements in cost, profitability, performance, efficiency, and utility of use in the field identified above. The Detailed Description includes an Introduction to facilitate understanding of the remainder of the Detailed Description. The Introduction includes Example Embodiments of one or more of systems, methods, articles of manufacture, and computer readable media in accordance with concepts described herein. As is discussed in more detail in the Conclusions, the invention encompasses all possible modifications and variations within the scope of the issued claims. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1A  illustrates selected details of an embodiment of a Solid-State Disk (SSD) including an SSD controller providing fractional higher-level redundancy for Non-Volatile Memories (NVMs). 
         FIG. 1B  illustrates selected details of various embodiments of systems including one or more instances of the SSD of  FIG. 1A . 
         FIG. 2  illustrates selected details of an embodiment of mapping a Logical Page Number (LPN) portion of a Logical Block Address (LBA). 
         FIG. 3  illustrates selected details of an embodiment of accessing a Non-Volatile Memory (NVM) at a read unit address to produce read data organized as various read units, collectively having a length measured in quanta of read units. 
         FIG. 4A  illustrates selected details of an embodiment of a read unit. 
         FIG. 4B  illustrates selected details of another embodiment of a read unit. 
         FIG. 5  illustrates selected details of an embodiment of a header having a number of fields. 
         FIG. 6  illustrates selected details of an embodiment of blocks, pages, and read units of multiple NVM devices (e.g. one or more flash die and/or flash chips) managed in logical slices and/or sections. 
         FIG. 7  illustrates selected details of various embodiments of higher-level redundancy techniques. 
         FIG. 8  illustrates selected details of an embodiment of dynamic higher-level redundancy mode management with a Redundant Array of Silicon Independent Elements (RASIE). 
         FIG. 9  illustrates an embodiment of read units having lower-level redundancy information of adaptive code rates protected by higher-level redundancy information stored in one or more of the read units. 
         FIG. 10  illustrates selected details of an embodiment of higher-level redundancy information result and data source correspondences. 
         FIG. 11  illustrates selected details of an embodiment of higher-level redundancy information computations. 
         FIG. 12  illustrates selected details of an embodiment of recovery from one (lower-level) failure (during a single operation). 
         FIGS. 13A-13D  illustrate selected details of an embodiment of recovery from two (lower-level) failures (during a single operation). 
         FIGS. 14A and 14B  illustrate selected details of an embodiment of computing higher-level redundancy information with respect to pages received from NVMs. 
         FIGS. 15A-15C  illustrate selected details of an embodiment of backing out of a computation of higher-level redundancy information with respect to a write provided to NVMs. 
         FIGS. 16A-16C  illustrate selected details of an embodiment of fractional higher-level redundancy. 
     
    
    
     LIST OF REFERENCE SYMBOLS IN DRAWINGS 
     
       
         
           
               
               
             
               
                   
               
               
                 Ref.  
                   
               
               
                 Symbol 
                 Element Name 
               
               
                   
               
             
            
               
                 100 
                 SSD Controller 
               
               
                 101 
                 SSD 
               
               
                 102 
                 Host 
               
               
                 103 
                 (optional) Switch/Fabric/Intermediate Controller 
               
               
                 104 
                 Intermediate Interfaces 
               
               
                 105 
                 OS 
               
               
                 106 
                 FirmWare (FW) 
               
               
                 107 
                 Driver 
               
               
                 107D 
                 dotted-arrow 
               
               
                   
                 (Host Software      I/O Device Communication) 
               
               
                 109 
                 Application 
               
               
                 109D 
                 dotted-arrow 
               
               
                   
                 (Application      I/O Device Communication via driver) 
               
               
                 109V 
                 dotted-arrow 
               
               
                   
                 (Application      I/O Device Communication via VF) 
               
               
                 110 
                 External Interfaces 
               
               
                 111 
                 Host Interfaces 
               
               
                 112C 
                 (optional) Card Memory 
               
               
                 113 
                 Tag Tracking 
               
               
                 114 
                 Multi-Device Management Software 
               
               
                 115 
                 Host Software 
               
               
                 116 
                 I/O Card 
               
               
                 117 
                 I/O &amp; Storage Devices/Resources 
               
               
                 118 
                 Servers 
               
               
                 119 
                 LAN/WAN 
               
               
                 121 
                 Data Processing 
               
               
                 123 
                 Engines 
               
               
                 131 
                 Buffer 
               
               
                 133 
                 DMA 
               
               
                 135 
                 ECC-X 
               
               
                 137 
                 Memory 
               
               
                 141 
                 Map 
               
               
                 143 
                 Table 
               
               
                 151 
                 Recycler 
               
               
                 161 
                 ECC 
               
               
                 171 
                 CPU 
               
               
                 172 
                 CPU Core 
               
               
                 173 
                 Command Management 
               
               
                 175 
                 Buffer Management 
               
               
                 177 
                 Translation Management 
               
               
                 179 
                 Coherency Management 
               
               
                 180 
                 Memory Interface 
               
               
                 181 
                 Device Management 
               
               
                 182 
                 Identity Management 
               
               
                 190 
                 Device Interfaces 
               
               
                 191 
                 Device Interface Logic 
               
               
                 192 
                 Flash Device 
               
               
                 193 
                 Scheduling 
               
               
                 194 
                 Flash Die 
               
               
                 199 
                 NVM 
               
               
                 211 
                 LBA 
               
               
                 213 
                 LPN 
               
               
                 215 
                 Logical Offset 
               
               
                 221 
                 Map Info for LPN 
               
               
                 223 
                 Read Unit Address 
               
               
                 225 
                 Length in Read Units 
               
               
                 311 
                 Read Data 
               
               
                 313 
                 First Read Unit 
               
               
                 315 
                 Last Read Unit 
               
               
                 401A 
                 Read Unit 
               
               
                 401B 
                 Read Unit 
               
               
                 410B 
                 Header Marker (HM) 
               
               
                 411A 
                 Header 1 
               
               
                 411B 
                 Header 1 
               
               
                 412B 
                 Header 2 
               
               
                 419A 
                 Header N 
               
               
                 419B 
                 Header N 
               
               
                 421A 
                 Data Bytes 
               
               
                 421B 
                 Data Bytes 
               
               
                 422B 
                 Data Bytes 
               
               
                 429B 
                 Data Bytes 
               
               
                 431A 
                 Optional Padding Bytes 
               
               
                 431B 
                 Optional Padding Bytes 
               
               
                 501 
                 Header 
               
               
                 511 
                 Type 
               
               
                 513 
                 Last Indicator 
               
               
                 515 
                 Flags 
               
               
                 517 
                 LPN 
               
               
                 519 
                 Length 
               
               
                 521 
                 Offset 
               
               
                 600 
                 Striping Direction 
               
               
                 610.0,  
                 Flash Die 
               
               
                 610.1, 
                   
               
               
                 610.61, 
                   
               
               
                 610.62, 
                   
               
               
                 610.63, 
                   
               
               
                 610.64, 
                   
               
               
                 610.65 
                   
               
               
                 610.0B0, 
                 Blocks 
               
               
                 610.0B1, 
                   
               
               
                 610.0B2, 
                   
               
               
                 610.0BB, 
                   
               
               
                 610.1B0, 
                   
               
               
                 610.1B1, 
                   
               
               
                 610.1B2, 
                   
               
               
                 610.1BB, 
                   
               
               
                 610.63B0, 
                   
               
               
                 610.63B1, 
                   
               
               
                 610.63B2, 
                   
               
               
                 610.63BB, 
                   
               
               
                 610.64B0, 
                   
               
               
                 610.64B1, 
                   
               
               
                 610.6482, 
                   
               
               
                 610.6488, 
                   
               
               
                 610.6580, 
                   
               
               
                 610.6581, 
                   
               
               
                 610.6582, 
                   
               
               
                 610.6588 
                   
               
               
                 610.0P0, 
                 Pages 
               
               
                 610.0P1, 
                   
               
               
                 610.0P2, 
                   
               
               
                 610.0PP, 
                   
               
               
                 610.63P0, 
                   
               
               
                 610.63P1, 
                   
               
               
                 610.63P2, 
                   
               
               
                 610.63PP, 
                   
               
               
                 610.64P0, 
                   
               
               
                 610.64P1, 
                   
               
               
                 610.64P2, 
                   
               
               
                 610.64PP, 
                   
               
               
                 610.65P0, 
                   
               
               
                 610.65P1, 
                   
               
               
                 610.65P2, 
                   
               
               
                 610.65PP 
                   
               
               
                 610.0R0, 
                 Read Units (RUs) 
               
               
                 610.0R1, 
                   
               
               
                 610.0RR, 
                   
               
               
                 610.1R0, 
                   
               
               
                 610.1R1, 
                   
               
               
                 610.1RR, 
                   
               
               
                 610.65R0, 
                   
               
               
                 610.65R1, 
                   
               
               
                 610.65RR 
                   
               
               
                 660.0,  
                 R-blocks 
               
               
                 660.1, 
                   
               
               
                 660.2, 
                   
               
               
                 660.R 
                   
               
               
                 720 
                 Flash Device (s) 
               
               
                 730 
                 Interface Channel (s) 
               
               
                 740 
                 Extra Flash Device (s) 
               
               
                 750 
                 Extra Interface Channel (s) 
               
               
                 801 
                 Start 
               
               
                 802 
                 Operate in First Higher-Level Redundancy Mode 
               
               
                 803 
                 Failure? 
               
               
                 804 
                 Reduce Free Space 
               
               
                 805 
                 Rearrange Data Storage 
               
               
                 806 
                 Recover/Store Failed User Data 
               
               
                 807 
                 Determine/Store Revised Higher-Level Redundancy 
               
               
                   
                 Information 
               
               
                 808 
                 Operate in Second Higher-Level Redundancy Mode 
               
               
                 809 
                 Dynamically Transition Operating Mode 
               
               
                 899 
                 End 
               
               
                 911, 
                 Read Unit 
               
               
                 931, 
                   
               
               
                 951, 
                   
               
               
                 971 
                   
               
               
                 911.E,  
                 Lower-Level ECC 
               
               
                 931.E, 
                   
               
               
                 951.E,  
                   
               
               
                 971.E 
                   
               
               
                 911.U, 
                 User Data 
               
               
                 931.U, 
                   
               
               
                 951.U,  
                   
               
               
                 971.U 
                   
               
               
                 1001 
                 Result 
               
               
                 1002 
                 Result 
               
               
                 1003 
                 Result 
               
               
                 1010 
                 R0 
               
               
                 1011 
                 R1 
               
               
                 1019 
                 Data 
               
               
                 1401A 
                 Start 
               
               
                 1401B 
                 Start 
               
               
                 1402A 
                 Issue Read Operation 
               
               
                 1402B 
                 Page Ready? 
               
               
                 1403A 
                 All Issued? 
               
               
                 1403B 
                 Perform Page-Based Computations 
               
               
                 1404B 
                 Pages Finished? 
               
               
                 1499A 
                 End 
               
               
                 1499B 
                 End 
               
               
                 1501A 
                 Start 
               
               
                 1501B 
                 Start 
               
               
                 1502A 
                 Issue Write Operation 
               
               
                 1502B 
                 Write OK? 
               
               
                 1503A 
                 All Issued? 
               
               
                 1503B 
                 Backout Write from Redundancy 
               
               
                 1504B 
                 Writes Finished? 
               
               
                 1599A 
                 End 
               
               
                 1599B 
                 End 
               
               
                 1600, 
                 R-pages 
               
               
                 1601, 
                   
               
               
                 1602, 
                   
               
               
                 1609 
                   
               
               
                 1600.D, 
                 Data Information 
               
               
                 1601.D, 
                   
               
               
                 1602.D, 
                   
               
               
                 1609.D, 
                   
               
               
                 1610.D, 
                   
               
               
                 1620.D, 
                   
               
               
                 1690.D 
                   
               
               
                 1600.R, 
                 Redundancy Information 
               
               
                 1601.R, 
                   
               
               
                 1602.R, 
                   
               
               
                 1609.R, 
                   
               
               
                 1610.R, 
                   
               
               
                 1620.R, 
                   
               
               
                 1690.R 
                   
               
               
                 1700 
                 Concurrently Operate in First and Second Granularity 
               
               
                   
                 Higher-Level Redundancy Modes 
               
               
                 1701 
                 Start 
               
               
                 1702 
                 Operate in First Granularity Higher-Level Redundancy Mode 
               
               
                 1703 
                 Read/Write According to First Granularity Higher-Level 
               
               
                   
                 Redundancy Mode 
               
               
                 1704 
                 Operate in Second Granularity Higher-Level Redundancy  
               
               
                   
                 Mode 
               
               
                 1705 
                 Read/Write According to Second Granularity Higher-Level  
               
               
                   
                 Redundancy Mode 
               
               
                 1706 
                 Granularity? 
               
               
                 1706.1 
                 First 
               
               
                 1706.2 
                 Second 
               
               
                 1707 
                 Read/Write Request 
               
               
                   
               
            
           
         
       
     
     DETAILED DESCRIPTION 
     A detailed description of one or more embodiments of the invention is provided below along with accompanying figures illustrating selected details of the invention. The invention is described in connection with the embodiments. The embodiments herein are understood to be merely exemplary, the invention is expressly not limited to or by any or all of the embodiments herein, and the invention encompasses numerous alternatives, modifications, and equivalents. To avoid monotony in the exposition, a variety of word labels (such as: first, last, certain, various, further, other, particular, select, some, and notable) may be applied to separate sets of embodiments; as used herein such labels are expressly not meant to convey quality, or any form of preference or prejudice, but merely to conveniently distinguish among the separate sets. The order of some operations of disclosed processes is alterable within the scope of the invention. Wherever multiple embodiments serve to describe variations in process, system, and/or program instruction features, other embodiments are contemplated that in accordance with a predetermined or a dynamically determined criterion perform static and/or dynamic selection of one of a plurality of modes of operation corresponding respectively to a plurality of the multiple embodiments. Numerous specific details are set forth in the following description to provide a thorough understanding of the invention. The details are provided for the purpose of example and the invention may be practiced according to the claims without some or all of the details. For the purpose of clarity, technical material that is known in the technical fields related to the invention has not been described in detail so that the invention is not unnecessarily obscured. 
     INTRODUCTION 
     This introduction is included only to facilitate the more rapid understanding of the Detailed Description; the invention is not limited to the concepts presented in the introduction (including explicit examples, if any), as the paragraphs of any introduction are necessarily an abridged view of the entire subject and are not meant to be an exhaustive or restrictive description. For example, the introduction that follows provides overview information limited by space and organization to only certain embodiments. There are many other embodiments, including those to which claims will ultimately be drawn, discussed throughout the balance of the specification. 
     Acronyms 
     At least some of the various shorthand abbreviations (e.g. acronyms) defined here refer to certain elements used herein. 
     
       
         
           
               
               
             
               
                   
               
               
                 Acronym 
                 Description 
               
               
                   
               
             
            
               
                 AHCI 
                 Advanced Host Controller Interface 
               
               
                 API 
                 Application Program Interface 
               
               
                 ATA 
                 Advanced Technology Attachment (AT Attachment) 
               
               
                 BCH 
                 Bose Chaudhuri Hocquenghem 
               
               
                 CD 
                 Compact Disk 
               
               
                 CF 
                 Compact Flash 
               
               
                 CMOS 
                 Complementary Metal Oxide Semiconductor 
               
               
                 CPU 
                 Central Processing Unit 
               
               
                 CRC 
                 Cyclic Redundancy Check 
               
               
                 DAS 
                 Direct Attached Storage 
               
               
                 DDR 
                 Double-Data-Rate 
               
               
                 DMA 
                 Direct Memory Access 
               
               
                 DNA 
                 Direct NAND Access 
               
               
                 DRAM 
                 Dynamic Random Access Memory 
               
               
                 DVD 
                 Digital Versatile/Video Disk 
               
               
                 DVR 
                 Digital Video Recorder 
               
               
                 ECC 
                 Error-Correcting Code 
               
               
                 eMMC 
                 embedded MultiMediaCard 
               
               
                 eSATA 
                 external Serial Advanced Technology Attachment 
               
               
                 GPS 
                 Global Positioning System 
               
               
                 HDD 
                 Hard Disk Drive 
               
               
                 I/O 
                 Input/Output 
               
               
                 IC 
                 Integrated Circuit 
               
               
                 IDE 
                 Integrated Drive Electronics 
               
               
                 JPEG 
                 Joint Photographic Experts Group 
               
               
                 LAN 
                 Local Area Network 
               
               
                 LBA 
                 Logical Block Address 
               
               
                 LDPC 
                 Low-Density Parity-Check 
               
               
                 LPN 
                 Logical Page Number 
               
               
                 MLC 
                 Multi-Level Cell 
               
               
                 MMC 
                 MultiMediaCard 
               
               
                 MPEG 
                 Moving Picture Experts Group 
               
               
                 NAS 
                 Network Attached Storage 
               
               
                 NCQ 
                 Native Command Queuing 
               
               
                 NVM 
                 Non-Volatile Memory 
               
               
                 ONA 
                 Optimized NAND Access 
               
               
                 ONFI 
                 Open NAND Flash Interface 
               
               
                 OS 
                 Operating System 
               
               
                 PC 
                 Personal Computer 
               
               
                 PCIe 
                 Peripheral Component Interconnect express (PCI express) 
               
               
                 PDA 
                 Personal Digital Assistant 
               
               
                 PHY 
                 PHYsical interface 
               
               
                 POS 
                 Point Of Sale 
               
               
                 RAID 
                 Redundant Array of Inexpensive/Independent Disks 
               
               
                 RASIE 
                 Redundant Array of Silicon Independent Elements 
               
               
                 ReRAM 
                 Resistive Random Access Memory 
               
               
                 RS 
                 Reed-Solomon 
               
               
                 SAN 
                 Storage Attached Network 
               
               
                 SAS 
                 Serial Attached Small Computer System Interface (Serial SCSI) 
               
               
                 SATA 
                 Serial Advanced Technology Attachment (Serial ATA) 
               
               
                 SCSI 
                 Small Computer System Interface 
               
               
                 SD 
                 Secure Digital 
               
               
                 SDR 
                 Single-Data-Rate 
               
               
                 SLC 
                 Single-Level Cell 
               
               
                 SMART 
                 Self-Monitoring Analysis and Reporting Technology 
               
               
                 SRAM 
                 Static Random Access Memory 
               
               
                 SSD 
                 Solid-State Disk/Drive 
               
               
                 UPS 
                 Unified Flash Storage 
               
               
                 USB 
                 Universal Serial Bus 
               
               
                 VF 
                 Virtual Function 
               
               
                 WAN 
                 Wide Area Network 
               
               
                   
               
            
           
         
       
     
     NAND flash memory uses an array of floating gate transistors to store information. In SLC technology, each bit cell (e.g. floating gate transistor) is enabled to store one bit of information. In MLC technology, each bit cell is enabled to store multiple bits of information. As manufacturing technology (e.g. CMOS technology) scales down, each floating gate stores fewer electrons. Further, as storage capacity and density increase, each bit cell stores more bits. Therefore, values stored in the bit cells are represented by smaller voltage ranges. Uncertainties in sensing and/or changes in amount of stored electrons over time increase a probability for data to be stored or read incorrectly. Use of one or more redundancy and/or ECC techniques (e.g. at a lower-level) enables correct retrieval of otherwise corrupted data from NAND flash memory, overcoming, in some usage scenarios, some of the aforementioned difficulties. 
     Some types of SSDs use flash memory to provide non-volatile storage (e.g., the flash memory retains information without application of power). Use of one or more ECC and/or redundancy techniques (e.g. at a higher-level) enables correct retrieval of otherwise corrupted data from flash memory, and/or enables proper system-level operation of an SSD even when one or more flash memory elements fail intermittently or permanently. 
     For example, an SSD controller enables dynamic higher-level redundancy mode management with independent silicon elements to provide graceful degradation as one or more NVM (e.g. flash) elements fail during operation of an SSD implemented in part by the controller. A portion of the NVM is read. If an error occurs that is not correctable using lower-level redundancy and/or error correction (such as in accordance with one or more ECC techniques), then higher-level redundancy and/or error correction (such as in accordance with one or more RASIE techniques and/or dynamic higher-level redundancy mode management techniques) is used to attempt to correct the error. If a failure of one of the NVM elements is detected by the lower-level and/or the higher-level redundancy and/or error correction, and/or by other techniques (such as a failing status reported by one or more of the NVM elements), then the higher-level redundancy and/or error correction is dynamically transitioned from operating in a current mode to operating in a new mode. The transition includes one or more of reducing free space available on the SSD, rearranging data storage of the SSD, recovering/storing failed user data (if possible), and determining/storing revised higher-level redundancy and/or error correction information. Operation then continues in the new mode. If another failure of another one of the NVM elements is detected with the higher-level redundancy and/or error correction now operating in the new mode, then another transition is made to another higher-level redundancy and/or error correction mode. Writing the NVM is in accordance with a higher-level redundancy and/or error correction operating mode, including determining/storing higher-level redundancy and/or error correction information in accordance with the higher-level redundancy and/or error correction operating mode and write data. 
     If greater than a threshold number and/or rate of lower-level error corrections and/or one or more failures occur for an area of memory, then optionally the area of memory is dynamically transitioned to operating in a new higher-level redundancy and/or error correction operating mode that is enabled to recover from more errors than a current higher-level redundancy and/or error correction operating mode. For example, if the current higher-level redundancy and/or error correction operating mode is none (e.g. no higher-level redundancy information is computed and/or used to recover from a failure of lower-level redundancy to correct an error), then the new higher-level redundancy and/or error correction operating mode is one that enables recovery from a single failure that is uncorrectable by the lower-level redundancy. For another example, if the current higher-level redundancy and/or error correction operating mode is one that enables recovery from a single failure that is uncorrectable by the lower-level redundancy, then the new higher-level redundancy and/or error correction operating mode is one that enables recovery from two failures that are uncorrectable by the lower-level redundancy. 
     Determining/storing higher-level redundancy and/or error correction information is in accordance, in some embodiments and/or usage scenarios, with computation of higher-level redundancy information. Higher-level redundancy information computation enables an SSD controller to provide higher-level redundancy capabilities to maintain reliable operation in a context of failures of non-volatile (e.g. flash) memory elements during operation of an SSD implemented in part by the controller. A first portion of higher-level redundancy information is computed using parity coding via an XOR of all pages in a portion of data (e.g. a stripe) to be protected by the higher-level redundancy information. A second portion of the higher-level redundancy information is computed using a weighted-sum technique, each page in the portion being assigned a unique non-zero “index” as a weight when computing the weighted-sum. Arithmetic is performed over a finite field (such as a Galois Field, or such as the integers mod p where p is a prime). 
     The portions of the higher-level redundancy information are computable in any order, such as an order determined by an order of completion of one or more read operations performed on NVM elements, or such as an order based on order of data returned and/or available from NVM elements, enabling, in various embodiments, reduced or eliminated buffering. The any order computability enables, in various embodiments, computing recovery data values and/or backing out of a write using relatively little temporary and/or intermediate buffering and/or state. The portions of the higher-level redundancy information are computable with any degree of parallelism, such as determined by available dedicated hardware elements, enabling, in various embodiments, reduced latency processing and/or reduced memory (e.g. NVM) bandwidth usage. 
     In some embodiments, one or more entire NVM elements (e.g. flash die) are dedicated to higher-level redundancy information, such as for various so-called “non-fractional” RASIE techniques. In some situations, customers of storage systems prefer storage capabilities at various so-called “binary capacity” points (for example 256 GB and 128 GB), and dedicating an entire NVM element or a plurality of entire NVM elements to error recovery results in burdensome increased costs. In some embodiments and/or usage scenarios, so-called “fractional” RASIE techniques reduce higher-level redundancy information overhead to less than one entire NVM element. The fractional higher-level redundancy techniques enable RASIE benefits for lower capacity points while retaining binary capacity. In some embodiments, fractional higher-level redundancy techniques include striping data across multiple blocks of NVM elements. 
     Example Embodiments 
     In concluding the introduction to the detailed description, what follows is a collection of example embodiments, including at least some explicitly enumerated as “ECs” (Example Combinations), providing additional description of a variety of embodiment types in accordance with the concepts described herein; these examples are not meant to be mutually exclusive, exhaustive, or restrictive; and the invention is not limited to these example embodiments but rather encompasses all possible modifications and variations within the scope of the issued claims and their equivalents. 
     EC1) A method comprising:
         writing a respective page to each of a respective first block of a plurality of non-volatile memory devices;   writing a respective page to each of a respective second block of the plurality of non-volatile memory devices, the respective second block different from the respective first block; and   wherein one or more of the respective pages contains redundancy of an erasure-correcting code protecting all of the respective pages.       

     EC2) The method of EC1, wherein a fraction of the respective pages containing the redundancy of the erasure-correcting code is less than 1 in a number of the plurality of non-volatile memory devices. 
     EC3) The method of EC2, wherein the fraction is 1 in an integer multiple of the number of the plurality of non-volatile memory devices. 
     EC4) The method of EC3, wherein the integer multiple is a power of two. 
     EC5) The method of EC1, wherein the erasure-correcting code is an error-correcting code. 
     EC6) The method of EC1, wherein the respective first blocks are dual-plane blocks. 
     EC7) A system comprising:
         a plurality of non-volatile memory devices, each of the non-volatile memory devices comprising a plurality of blocks, each of the blocks comprising a plurality of pages;   an erasure-correcting code generator operable to generate one or more pages of redundant data protecting a respective plurality of pages of user data, the one or more pages of redundant data and the respective plurality of pages of user data forming a codeword of an erasure-correcting code;   wherein the one or more pages of redundant data and the respective plurality of pages of user data are each enabled to be stored into a separate respective one of the pages of at least some of the blocks such that more than one of the non-volatile memory devices contains more than one of the one or more pages of redundant data and the respective plurality of pages of user data, and such that none of the at least some of the blocks contains more than one of the one or more pages of redundant data and the respective plurality of pages of user data; and   whereby a failure of at least one of the at least some of the blocks is correctable by the erasure-correcting code.       

     EC8) The system of EC7, wherein a number of the one or more pages of redundant data is equal to a number of the at least one of the at least some of the blocks that is correctable. 
     EC9) The system of EC7, further comprising an erasure-correcting code corrector enabled to process information read from at least some of the separate respective ones of the pages of the at least some of the blocks to correct others of the separate respective ones of the pages of the at least some of the blocks. 
     EC10) The system of EC7, further comprising an error-correcting code generator enabled to encode each of the one or more pages of redundant data and the respective plurality of pages of user data to produce respective error-correcting information, the respective error-correcting information enabled to be stored into the separate respective ones of the pages of the at least some of the blocks along with the corresponding one or more pages of redundant data and the respective plurality of pages of user data. 
     EC11) A system comprising:
         a means for computing one or more units of higher-level redundancy information based at least in part on a plurality of units of data storage information; and   wherein the means for computing comprises a means for accumulating a weighted sum of a respective non-zero unique constant value for each of the units of data storage information multiplied by contents of the units of data storage information as at least a portion of the units of higher-level redundancy information.       

     EC12) The system of EC11, further comprising a means for storing the units of higher-level redundancy information and the units of data storage information in portions of one or more non-volatile memory devices. 
     EC13) The system of EC12, wherein the portions comprise one or more integral numbers of pages of the non-volatile memory devices, and the non-volatile memory devices comprise one or more flash memories. 
     EC14) The system of EC11, wherein the units correspond to one or more integral numbers of pages of one or more flash memories. 
     EC15) The system of EC11, wherein the units of higher-level redundancy information are not computable as a remainder of a polynomial division by a generator polynomial of corresponding bytes of the units of data storage information. 
     EC16) The system of EC11, wherein the means for accumulating is enabled to accumulate at least a portion of the weighted sum incrementally. 
     EC17) The system of EC16, wherein the means for accumulating is further enabled to process more than one of the units of data storage information in parallel. 
     EC18) The system of EC17, wherein the more than one of the units correspond to one or more integral numbers of pages of one or more flash memories. 
     EC19) The system of EC11, wherein the means for accumulating is enabled to accumulate at least a portion of the weighted sum in an ordering corresponding to an ordering of read operation completion by one or more flash memories. 
     EC20) The system of EC11, wherein the means for accumulating is enabled to accumulate at least a portion of the weighted sum in an ordering corresponding to an ordering of data returned from one or more flash memories. 
     EC21) The system of EC20, wherein the ordering of data returned is based at least in part on an order that data is available from the one or more flash memories. 
     EC22) The system of EC11, wherein the means for accumulating is enabled to accumulate at least a portion of the weighted sum in parallel. 
     EC23) The system of EC11, wherein the means for accumulating is enabled to accumulate at least a portion of the weighted sum in parallel, the portion corresponding to elements of the units of data storage information that are retrievable from corresponding pages of one or more flash memories. 
     EC24) The system of EC23, wherein the elements are determined, at least in part, by an order of completion of read operations of the corresponding pages. 
     EC25) The system of EC12, further comprising a means for determining if one or more of the portions have a lower-level error correction failure when read. 
     EC26) The system of EC12, further comprising a means for reading one or more of the portions. 
     EC27) The system of EC11, wherein the means for accumulating a weighted sum is enabled to selectively exclude from the weighted sum up to two of the units of data storage information. 
     EC28) The system of EC27, further comprising means for processing results of the means for accumulating to restore the excluded units of the data storage information. 
     EC29) The system of EC27, wherein the means for accumulating is enabled to accumulate at least a portion of the weighted sum incrementally and in an ordering corresponding to an ordering of read operation completion by one or more non-volatile memory devices. 
     EC30) The system of EC27, wherein the means for accumulating is enabled to accumulate at least a portion of the weighted sum incrementally and in an ordering corresponding to an ordering of data returned from one or more non-volatile memory devices. 
     EC31) The system of EC30, wherein the ordering of data returned is based at least in part on an order that data is available from the one or more non-volatile memory devices. 
     EC32) The system of EC11, wherein the units of higher-level redundancy information and the units of data storage information correspond to respective pages of one or more flash memories. 
     EC33) The system of EC32, wherein the flash memories are comprised of a plurality of die, and each of the respective pages are on a unique one of the die. 
     EC34) The system of EC11, further comprising a means for storing the units of higher-level redundancy information and the units of data storage information in portions of one or more flash memories at least in part in response to requests from a computing host. 
     EC35) The system of EC34, further comprising a means for interfacing the requests with the computing host. 
     EC36) The system of EC35, wherein the means for interfacing the requests with the computing host is compatible with a storage interface standard. 
     EC37) The system of EC34, wherein the means for storing comprises a means for interfacing with the flash memories. 
     EC38) The system of EC37, wherein the means for interfacing with the flash memories comprises a flash memory interface. 
     EC39) The system of EC34, further comprising:
         a means for interfacing the requests with the computing host; and   wherein the means for storing comprises a means for interfacing with the flash memories.       

     EC40) The system of EC39, wherein the means are collectively implemented in a single Integrated Circuit (IC). 
     EC41) The system of EC39, wherein the means are comprised in a Solid-State Disk (SSD). 
     EC42) The system of EC34, further comprising all or any portions of the computing host. 
     EC43) The system of EC13, further comprising at least one of the flash memories. 
     EC44) A method comprising:
         computing one or more units of higher-level redundancy information based at least in part on a plurality of units of data storage information; and   wherein the computing comprises accumulating a weighted sum of a respective non-zero unique constant value for each of the units of data storage information multiplied by contents of the units of data storage information as at least a portion of the units of higher-level redundancy information.       

     EC45) The method of EC44, further comprising storing the units of higher-level redundancy information and the units of data storage information in portions of one or more non-volatile memory devices. 
     EC46) The method of EC45, wherein the portions comprise one or more integral numbers of pages of the non-volatile memory devices, and the non-volatile memory devices comprise one or more flash memories. 
     EC47) The method of EC44, wherein the units correspond to one or more integral numbers of pages of one or more flash memories. 
     EC48) The method of EC44, wherein the units of higher-level redundancy information are not computable as a remainder of a polynomial division by a generator polynomial of corresponding bytes of the units of data storage information. 
     EC49) The method of EC44, wherein the accumulating comprises accumulating at least a portion of the weighted sum incrementally. 
     EC50) The method of EC49, wherein the accumulating further comprises processing more than one of the units of data storage information in parallel. 
     EC51) The method of EC50, wherein the more than one of the units correspond to one or more integral numbers of pages of one or more flash memories. 
     EC52) The method of EC44, wherein the accumulating comprises accumulating at least a portion of the weighted sum in an ordering corresponding to an ordering of read operation completion by one or more flash memories. 
     EC53) The method of EC44, wherein the accumulating comprises accumulating at least a portion of the weighted sum in an ordering corresponding to an ordering of data returned from one or more flash memories. 
     EC54) The method of EC53, wherein the ordering of data returned is based at least in part on an order that data is available from the one or more flash memories. 
     EC55) The method of EC44, wherein the accumulating comprises accumulating at least a portion of the weighted sum in parallel. 
     EC56) The method of EC44, wherein the accumulating comprises accumulating at least a portion of the weighted sum in parallel, the portion corresponding to elements of the units of data storage information that are retrievable from corresponding pages of one or more flash memories. 
     EC57) The method of EC56, wherein the elements are determined, at least in part, by an order of completion of read operations of the corresponding pages. 
     EC58) The method of EC45, further comprising determining if one or more of the portions have a lower-level error correction failure when read. 
     EC59) The method of EC45, further comprising reading one or more of the portions. 
     EC60) The method of EC44, wherein the accumulating a weighted sum is selectively excludes from the weighted sum up to two of the units of data storage information. 
     EC61) The method of EC60, further comprising processing results of the accumulating to restore the excluded units of the data storage information. 
     EC62) The method of EC60, wherein the accumulating comprises accumulating at least a portion of the weighted sum incrementally and in an ordering corresponding to an ordering of read operation completion by one or more non-volatile memory devices. 
     EC63) The method of EC60, wherein the accumulating comprises accumulating at least a portion of the weighted sum incrementally and in an ordering corresponding to an ordering of data returned from one or more non-volatile memory devices. 
     EC64) The method of EC63, wherein the ordering of data returned is based at least in part on an order that data is available from the one or more non-volatile memory devices. 
     EC65) The method of EC44, wherein the units of higher-level redundancy information and the units of data storage information correspond to respective pages of one or more flash memories. 
     EC66) The method of EC65, wherein the flash memories are comprised of a plurality of die, and each of the respective pages are on a unique one of the die. 
     EC67) The method of EC44, further comprising storing the units of higher-level redundancy information and the units of data storage information in portions of one or more flash memories at least in part in response to requests from a computing host. 
     EC68) The method of EC67, further comprising interfacing the requests with the computing host. 
     EC69) The method of EC68, wherein the interfacing the requests with the computing host is compatible with a storage interface standard. 
     EC70) The method of EC67, wherein the storing comprises interfacing with the flash memories. 
     EC71) The method of EC70, wherein the interfacing with the flash memories comprises a flash memory interface. 
     EC72) The method of EC67, further comprising:
         interfacing the requests with the computing host at least in part via computing host interface logic circuitry; and   wherein the storing is at least in part via flash memory interface logic circuitry enabled to interface with the flash memories.       

     EC73) The method of EC72, wherein the computing host interface logic circuitry and the flash memory interface logic circuitry are collectively implemented in a single Integrated Circuit (IC). 
     EC74) The method of EC72, wherein the computing host interface logic circuitry and the flash memory interface logic circuitry are comprised in a Solid-State Disk (SSD). 
     EC75) The method of EC67, further comprising operating all or any portions of the computing host. 
     EC76) The method of EC46, further comprising operating at least one of the flash memories. 
     EC77) A system comprising:
         computing logic circuitry enabled to compute one or more units of higher-level redundancy information based at least in part on a plurality of units of data storage information; and   wherein the computing logic circuitry comprises accumulating logic circuitry enabled to accumulate a weighted sum of a respective non-zero unique constant value for each of the units of data storage information multiplied by contents of the units of data storage information as at least a portion of the units of higher-level redundancy information.       

     EC78) The system of EC77, further comprising logic circuitry enabled to store the units of higher-level redundancy information and the units of data storage information in portions of one or more non-volatile memory devices. 
     EC79) The system of EC78, wherein the portions comprise one or more integral numbers of pages of the non-volatile memory devices, and the non-volatile memory devices comprise one or more flash memories. 
     EC80) The system of EC77, wherein the units correspond to one or more integral numbers of pages of one or more flash memories. 
     EC81) The system of EC77, wherein the units of higher-level redundancy information are not computable as a remainder of a polynomial division by a generator polynomial of corresponding bytes of the units of data storage information. 
     EC82) The system of EC77, wherein the accumulating logic circuitry is further enabled to accumulate at least a portion of the weighted sum incrementally. 
     EC83) The system of EC82, wherein the accumulating logic circuitry is further enabled to process more than one of the units of data storage information in parallel. 
     EC84) The system of EC83, wherein the more than one of the units correspond to one or more integral numbers of pages of one or more flash memories. 
     EC85) The system of EC77, wherein the accumulating logic circuitry is further enabled to accumulate at least a portion of the weighted sum in an ordering corresponding to an ordering of read operation completion by one or more flash memories. 
     EC86) The system of EC77, wherein the accumulating logic circuitry is further enabled to accumulate at least a portion of the weighted sum in an ordering corresponding to an ordering of data returned from one or more flash memories. 
     EC87) The system of EC86, wherein the ordering of data returned is based at least in part on an order that data is available from the one or more flash memories. 
     EC88) The system of EC77, wherein the accumulating logic circuitry is further enabled to accumulate at least a portion of the weighted sum in parallel. 
     EC89) The system of EC77, wherein the accumulating logic circuitry is further enabled to accumulate at least a portion of the weighted sum in parallel, the portion corresponding to elements of the units of data storage information that are retrievable from corresponding pages of one or more flash memories. 
     EC90) The system of EC89, wherein the elements are determined, at least in part, by an order of completion of read operations of the corresponding pages. 
     EC91) The system of EC78, further comprising logic circuitry enabled to determine if one or more of the portions have a lower-level error correction failure when read. 
     EC92) The system of EC78, further comprising logic circuitry enabled to read one or more of the portions. 
     EC93) The system of EC77, wherein the accumulating logic circuitry is further enabled to selectively exclude from the weighted sum up to two of the units of data storage information. 
     EC94) The system of EC93, further comprising logic circuitry enabled to process results of the accumulating logic circuitry to restore the excluded units of the data storage information. 
     EC95) The system of EC93, wherein the accumulating logic circuitry is further enabled to accumulate at least a portion of the weighted sum incrementally and in an ordering corresponding to an ordering of read operation completion by one or more non-volatile memory devices. 
     EC96) The system of EC93, wherein the accumulating logic circuitry is further enabled to accumulate at least a portion of the weighted sum incrementally and in an ordering corresponding to an ordering of data returned from one or more non-volatile memory devices. 
     EC97) The system of EC96, wherein the ordering of data returned is based at least in part on an order that data is available from the one or more non-volatile memory devices. 
     EC98) The system of EC77, wherein the units of higher-level redundancy information and the units of data storage information correspond to respective pages of one or more flash memories. 
     EC99) The system of EC98, wherein the flash memories are comprised of a plurality of die, and each of the respective pages are on a unique one of the die. 
     EC100) The system of EC77, further comprising flash storage logic circuitry enabled to store the units of higher-level redundancy information and the units of data storage information in portions of one or more flash memories at least in part in response to requests from a computing host. 
     EC101) The system of EC100, further comprising computing host interface logic circuitry enabled to interface the requests with the computing host. 
     EC102) The system of EC101, wherein the computing host interface logic circuitry is compatible with a storage interface standard. 
     EC103) The system of EC100, wherein the flash storage logic circuitry comprises flash memory interface logic circuitry enabled to interface with the flash memories. 
     EC104) The system of EC103, wherein the flash memory interface logic circuitry comprises a flash memory interface. 
     EC105) The system of EC100, further comprising:
         computing host interface logic circuitry enabled to interface the requests with the computing host; and   wherein the flash storage logic circuitry comprises flash memory interface logic circuitry enabled to interface with the flash memories.       

     EC106) The system of EC105, wherein the computing host interface logic circuitry and the flash memory interface logic circuitry are collectively implemented in a single Integrated Circuit (IC). 
     EC107) The system of EC105, wherein the computing host interface logic circuitry and the flash memory interface logic circuitry are comprised in a Solid-State Disk (SSD). 
     EC108) The system of EC100, further comprising all or any portions of the computing host. 
     EC109) The system of EC79, further comprising at least one of the flash memories. 
     EC110) A tangible computer readable medium having a set of instructions stored therein that when executed by a processing element cause the processing element to perform operations comprising:
         managing computing one or more units of higher-level redundancy information based at least in part on a plurality of units of data storage information; and   wherein the computing comprises accumulating a weighted sum of a respective non-zero unique constant value for each of the units of data storage information multiplied by contents of the units of data storage information as at least a portion of the units of higher-level redundancy information.       

     EC111) The tangible computer readable medium of EC110, wherein the operations further comprise managing storing the units of higher-level redundancy information and the units of data storage information in portions of one or more non-volatile memory devices. 
     EC112) The tangible computer readable medium of EC111, wherein the portions comprise one or more integral numbers of pages of the non-volatile memory devices, and the non-volatile memory devices comprise one or more flash memories. 
     EC113) The tangible computer readable medium of EC110, wherein the units correspond to one or more integral numbers of pages of one or more flash memories. 
     EC114) The tangible computer readable medium of EC110, wherein the units of higher-level redundancy information are not computable as a remainder of a polynomial division by a generator polynomial of corresponding bytes of the units of data storage information. 
     EC115) The tangible computer readable medium of EC110, wherein the accumulating comprises accumulating at least a portion of the weighted sum incrementally. 
     EC116) The tangible computer readable medium of EC115, wherein the accumulating further comprises processing more than one of the units of data storage information in parallel. 
     EC117) The tangible computer readable medium of EC116, wherein the more than one of the units correspond to one or more integral numbers of pages of one or more flash memories. 
     EC118) The tangible computer readable medium of EC110, wherein the accumulating comprises accumulating at least a portion of the weighted sum in an ordering corresponding to an ordering of read operation completion by one or more flash memories. 
     EC119) The tangible computer readable medium of EC110, wherein the accumulating comprises accumulating at least a portion of the weighted sum in an ordering corresponding to an ordering of data returned from one or more flash memories. 
     EC120) The tangible computer readable medium of EC119, wherein the ordering of data returned is based at least in part on an order that data is available from the one or more flash memories. 
     EC121) The tangible computer readable medium of EC110, wherein the accumulating comprises accumulating at least a portion of the weighted sum in parallel. 
     EC122) The tangible computer readable medium of EC110, wherein the accumulating comprises accumulating at least a portion of the weighted sum in parallel, the portion corresponding to elements of the units of data storage information that are retrievable from corresponding pages of one or more flash memories. 
     EC123) The tangible computer readable medium of EC122, wherein the elements are determined, at least in part, by an order of completion of read operations of the corresponding pages. 
     EC124) The tangible computer readable medium of EC111, wherein the operations further comprise managing determining if one or more of the portions have a lower-level error correction failure when read. 
     EC125) The tangible computer readable medium of EC111, wherein the operations further comprise managing reading one or more of the portions. 
     EC126) The tangible computer readable medium of EC110, wherein the accumulating a weighted sum is selectively excludes from the weighted sum up to two of the units of data storage information. 
     EC127) The tangible computer readable medium of EC126, wherein the operations further comprise managing processing results of the accumulating to restore the excluded units of the data storage information. 
     EC128) The tangible computer readable medium of EC126, wherein the accumulating comprises accumulating at least a portion of the weighted sum incrementally and in an ordering corresponding to an ordering of read operation completion by one or more non-volatile memory devices. 
     EC129) The tangible computer readable medium of EC126, wherein the accumulating comprises accumulating at least a portion of the weighted sum incrementally and in an ordering corresponding to an ordering of data returned from one or more non-volatile memory devices. 
     EC130) The tangible computer readable medium of EC129, wherein the ordering of data returned is based at least in part on an order that data is available from the one or more non-volatile memory devices. 
     EC131) The tangible computer readable medium of EC110, wherein the units of higher-level redundancy information and the units of data storage information correspond to respective pages of one or more flash memories. 
     EC132) The tangible computer readable medium of EC131, wherein the flash memories are comprised of a plurality of die, and each of the respective pages are on a unique one of the die. 
     EC133) The tangible computer readable medium of EC110, wherein the operations further comprise managing storing the units of higher-level redundancy information and the units of data storage information in portions of one or more flash memories at least in part in response to requests from a computing host. 
     EC134) The tangible computer readable medium of EC133, wherein the operations further comprise managing interfacing the requests with the computing host. 
     EC135) The tangible computer readable medium of EC134, wherein the interfacing the requests with the computing host is compatible with a storage interface standard. 
     EC136) The tangible computer readable medium of EC133, wherein the storing comprises interfacing with the flash memories. 
     EC137) The tangible computer readable medium of EC136, wherein the interfacing with the flash memories comprises a flash memory interface. 
     EC138) The tangible computer readable medium of EC133, wherein the operations further comprise:
         managing interfacing the requests with the computing host at least in part via managing computing host interface logic circuitry; and   wherein the storing is at least in part via flash memory interface logic circuitry enabled to interface with the flash memories.       

     EC139) The tangible computer readable medium of EC138, wherein the computing host interface logic circuitry and the flash memory interface logic circuitry are collectively implemented in a single Integrated Circuit (IC). 
     EC140) The tangible computer readable medium of EC138, wherein the computing host interface logic circuitry and the flash memory interface logic circuitry are comprised in a Solid-State Disk (SSD). 
     EC141) The tangible computer readable medium of EC133, wherein the operations further comprise managing operating all or any portions of the computing host. 
     EC142) The tangible computer readable medium of EC112, wherein the operations further comprise managing operating at least one of the flash memories. 
     EC143) A method comprising:
         computing one or more pages of higher-level redundancy information based at least in part on a plurality of pages of data storage information;   storing the pages of higher-level redundancy information and the pages of data storage information in pages of one or more flash memories; and   wherein the computing comprises accumulating a weighted sum of a respective non-zero unique constant value for each of the pages of data storage information multiplied by contents of the pages of data storage information as at least a portion of the pages of higher-level redundancy information.       

     EC144) The method of EC143, wherein the pages of higher-level redundancy information are not computable as a remainder of a polynomial division by a generator polynomial of corresponding bytes of the pages of data storage information. 
     EC145) The method of EC143, wherein the accumulating comprises accumulating incrementally. 
     EC146) The method of EC145, wherein the accumulating further comprises processing at least partially in parallel more than one of the pages of data storage information. 
     EC147) The method of EC143, further comprising reading at least some pages stored in the flash memories; and determining if any of the pages read are uncorrectable via lower-level redundancy information. 
     EC148) The method of EC143, further comprising computing a correction version of the higher-level redundancy information, wherein the computing a correction version of the higher-level redundancy information selectively excludes up to two pages of the pages of data storage information. 
     EC149) The method of EC148, further comprising processing results of the computing a correction version of the higher-level redundancy information to restore the excluded pages of the data storage information. 
     EC150) The method of EC148, wherein the accumulating comprises accumulating incrementally at least partially in an order determined at least in part by an order that read operations are completed by the flash memories. 
     EC151) The method of EC143, wherein the flash memories are comprised of a plurality of die, and only one of the pages of the higher-level redundancy information or the data storage information is stored in any one of the die. 
     EC152) The method of EC151, wherein the pages of the higher-level redundancy information are excluded from at least one of the die. 
     EC153) The method of EC151, wherein the pages of the data storage information are excluded from at least one of the die. 
     EC154) Any of the foregoing ECs having or referring to a storage interface standard, wherein the storage interface standard comprises one or more of
         a Universal Serial Bus (USB) interface standard,   a Compact Flash (CF) interface standard,   a MultiMediaCard (MMC) interface standard,   an embedded MMC (eMMC) interface standard,   a Thunderbolt interface standard,   a UFS interface standard,   a Secure Digital (SD) interface standard,   a Memory Stick interface standard,   an xD-picture card interface standard,   an Integrated Drive Electronics (IDE) interface standard,   a Serial Advanced Technology Attachment (SATA) interface standard,   an external SATA (eSATA) interface standard,   a Small Computer System Interface (SCSI) interface standard,   a Serial Attached Small Computer System Interface (SAS) interface standard,   a Fibre Channel interface standard,   an Ethernet interface standard, and   a Peripheral Component Interconnect express (PCIe) interface standard.       

     EC155) Any of the foregoing ECs having or referring to a flash memory interface, wherein the flash memory interface is compatible with one or more of
         an Open NAND Flash Interface (ONFI),   a Toggle-mode interface,   a Double-Data-Rate (DDR) synchronous interface,   a DDR2 synchronous interface;   a synchronous interface, and an asynchronous interface.       

     EC156) Any of the foregoing ECs having or referring to a computing host, wherein the computing host comprises one or more of
         a computer,   a workstation computer,   a server computer,   a storage server,   a Storage Attached Network (SAN),   a Network Attached Storage (NAS) device,   a Direct Attached Storage (DAS) device,   a storage appliance,   a Personal Computer (PC),   a laptop computer,   a notebook computer,   a netbook computer,   a tablet device or computer,   an ultrabook computer,   an electronic reading device (an e-reader),   a Personal Digital Assistant (PDA),   a navigation system,   a (handheld) Global Positioning System (GPS) device,   an automotive control system,   an automotive media control system or computer,   a printer, copier or fax machine or all-in-one device,   a Point Of Sale POS device,   a cash-register,   a media player,   a television,   a media recorder,   a Digital Video Recorder (DVR),   a digital camera,   a cellular handset,   a cordless telephone handset, and   an electronic game.       

     EC157) Any of the foregoing ECs having or referring to at least one flash memory, wherein at least a portion of the at least one flash memory comprises one or more of
         NAND flash technology storage cells, and   NOR flash technology storage cells.       

     EC158) Any of the foregoing ECs having or referring to at least one flash memory, wherein at least a portion of the at least one flash memory comprises one or more of
         Single-Level Cell (SLC) flash technology storage cells, and   Multi-Level Cell (MLC) flash technology storage cells.       

     EC159) Any of the foregoing ECs having or referring to at least one flash memory, wherein at least a portion of the at least one flash memory comprises one or more of
         polysilicon technology-based charge storage cells, and   silicon nitride technology-based charge storage cells.       

     EC160) Any of the foregoing ECs having or referring to at least one flash memory, wherein at least a portion of the at least one flash memory comprises one or more of
         two-dimensional technology-based flash memory technology, and   three-dimensional technology-based flash memory technology.
 
System
       

     In some embodiments, an I/O device, such as an SSD, includes an SSD controller. The SSD controller acts as a bridge between the host interface and NVM of the SSD, and executes commands of a host protocol sent from a computing host via a host interface of the SSD. At least some of the commands direct the SSD to write and read the NVM with data sent from and to the computing host, respectively. In further embodiments, the SSD controller is enabled to use a map to translate between LBAs of the host protocol and physical storage addresses in the NVM. In further embodiments, at least a portion of the map is used for private storage (not visible to the computing host) of the I/O device. For example, a portion of the LBAs not accessible by the computing host is used by the I/O device to manage access to logs, statistics, or other private data. 
     In some embodiments, accessing compressed data of varying-sized quanta in NVM provides improved storage efficiency in some usage scenarios. For example, an SSD controller receives (uncompressed) data from a computing host (e.g., relating to a disk write command), compresses the data, and stores the compressed data into flash memory. In response to a subsequent request from the computing host (e.g., relating to a disk read command), the SSD controller reads the compressed data from the flash memory, uncompresses the compressed data, and provides the uncompressed data to the computing host. The compressed data is stored in the flash memory according to varying-sized quanta, the quanta size varying due to, e.g., compression algorithm, operating mode, and compression effectiveness on various data. The SSD controller uncompresses the data in part by consulting an included map table to determine where header(s) are stored in the flash memory. The SSD controller parses the header(s) obtained from the flash memory to determine where appropriate (compressed) data is stored in the flash memory. The SSD controller uncompresses the appropriate data from the flash memory to produce the uncompressed data to provide to the computing host. In the instant application, uncompress (and variants thereof) is synonymous with decompress (and variants thereof). 
     In various embodiments, an SSD controller includes a host interface for interfacing with a computing host, an interface for interfacing with NVM such as flash memory, and circuitry for controlling the interfaces and performing (and/or controlling various aspects of the performing) compressing and uncompressing, as well as lower-level redundancy and/or error correction, higher-level redundancy and/or error correction, and dynamic higher-level redundancy mode management with independent silicon elements. 
     According to various embodiments, some host interfaces are compatible with one or more of a USB interface standard, a CF interface standard, an MMC interface standard, an eMMC interface standard, a Thunderbolt interface standard, a UFS interface standard, an SD interface standard, a Memory Stick interface standard, an xD-picture card interface standard, an IDE interface standard, a SATA interface standard, a SCSI interface standard, a SAS interface standard, and a PCIe interface standard. According to various embodiments, the computing host is all or any portions of a computer, a workstation computer, a server computer, a storage server, a SAN, a NAS device, a DAS device, a storage appliance, a PC, a laptop computer, a notebook computer, a netbook computer, a tablet device or computer, an ultrabook computer, an electronic reading device (such as an e-reader), a PDA, a navigation system, a (handheld) GPS device, an automotive control system, an automotive media control system or computer, a printer, copier or fax machine or all-in-one device, a POS device, a cash-register, a media player, a television, a media recorder, a DVR, a digital camera, a cellular handset, a cordless telephone handset, and an electronic game. In some embodiments, an interfacing host (such as a SAS/SATA bridge) operates as a computing host and/or as a bridge to a computing host. 
     In various embodiments, the SSD controller includes one or more processors. The processors execute firmware to control and/or perform operation of the SSD controller. The SSD controller communicates with the computing host to send and receive commands and/or status as well as data. The computing host executes one or more of an operating system, a driver, and an application. Communication by the computing host with the SSD controller is optionally and/or selectively via the driver and/or via the application. In a first example, all communication to the SSD controller is via the driver, and the application provides higher-level commands to the driver that the driver translates into specific commands for the SSD controller. In a second example, the driver implements a bypass mode and the application is enabled to send specific commands to the SSD controller via the driver. In a third example, a PCIe SSD controller supports one or more Virtual Functions (VFs), enabling an application, once configured, to communicate directly with the SSD controller, bypassing the driver. 
     According to various embodiments, some SSDs are compatible with form-factors, electrical interfaces, and/or protocols used by magnetic and/or optical non-volatile storage, such as HDDs, CD drives, and DVD drives. In various embodiments, SSDs use various combinations of zero or more parity codes, zero or more RS codes, zero or more BCH codes, zero or more Viterbi or other trellis codes, and zero or more LDPC codes. 
       FIG. 1A  illustrates selected details of an embodiment of an SSD including an SSD controller providing fractional higher-level redundancy for NVMs (e.g. flash memories, such as NAND flash memories). The SSD controller is for managing non-volatile storage, such as implemented via NVM elements (e.g., flash memories). SSD Controller  100  is communicatively coupled via one or more External Interfaces  110  to a host (not illustrated). According to various embodiments, External Interfaces  110  are one or more of: a SATA interface; a SAS interface; a PCIe interface; a Fibre Channel interface; an Ethernet Interface (such as 10 Gigabit Ethernet); a non-standard version of any of the preceding interfaces; a custom interface; or any other type of interface used to interconnect storage and/or communications and/or computing devices. For example, in some embodiments, SSD Controller  100  includes a SATA interface and a PCIe interface. 
     SSD Controller  100  is further communicatively coupled via one or more Device Interfaces  190  to NVM  199  including one or more storage devices, such as one or more instances of Flash Device  192 . According to various embodiments, Device Interfaces  190  are one or more of: an asynchronous interface; a synchronous interface; a single-data-rate (SDR) interface; a double-data-rate (DDR) interface; a DRAM-compatible DDR or DDR2 synchronous interface; an ONFI compatible interface, such as an ONFI  2 . 2  or ONFI  3 . 0  compatible interface; a Toggle-mode compatible flash interface; a non-standard version of any of the preceding interfaces; a custom interface; or any other type of interface used to connect to storage devices. 
     Each of Flash Device  192  has, in some embodiments, one or more individual Flash Die  194 . According to type of a particular one of Flash Device  192 , a plurality of Flash Die  194  in the particular Flash Device  192  is optionally and/or selectively accessible in parallel. Flash Device  192  is merely representative of one type of storage device enabled to communicatively couple to SSD Controller  100 . In various embodiments, any type of storage device is usable, such as an SLC NAND flash memory, MLC NAND flash memory, NOR flash memory, flash memory using polysilicon or silicon nitride technology-based charge storage cells, two- or three-dimensional technology-based flash memory, read-only memory, static random access memory, dynamic random access memory, ferromagnetic memory, phase-change memory, racetrack memory, ReRAM, or any other type of memory device or storage medium. 
     According to various embodiments, Device Interfaces  190  are organized as: one or more busses with one or more instances of Flash Device  192  per bus; one or more groups of busses with one or more instances of Flash Device  192  per bus, having busses in a group generally accessed in parallel; or any other organization of one or more instances of Flash Device  192  onto Device Interfaces  190 . 
     Continuing in  FIG. 1A , SSD Controller  100  has one or more modules, such as Host Interfaces  111 , Data Processing  121 , Buffer  131 , Map  141 , Recycler  151 , ECC  161 , Device Interface Logic  191 , and CPU  171 . The specific modules and interconnections illustrated in  FIG. 1A  are merely representative of one embodiment, and many arrangements and interconnections of some or all of the modules, as well as additional modules not illustrated, are conceived. In a first example, in some embodiments, there are two or more Host Interfaces  111  to provide dual-porting. In a second example, in some embodiments, Data Processing  121  and/or ECC  161  are combined with Buffer  131 . In a third example, in some embodiments, Host Interfaces  111  is directly coupled to Buffer  131 , and Data Processing  121  optionally and/or selectively operates on data stored in Buffer  131 . In a fourth example, in some embodiments, Device Interface Logic  191  is directly coupled to Buffer  131 , and ECC  161  optionally and/or selectively operates on data stored in Buffer  131 . 
     Host Interfaces  111  sends and receives commands and/or data via External Interfaces  110 , and, in some embodiments, tracks progress of individual commands via Tag Tracking  113 . For example, the commands include a read command specifying an address (such as an LBA) and an amount of data (such as a number of LBA quanta, e.g., sectors) to read; in response the SSD provides read status and/or read data. For another example, the commands include a write command specifying an address (such as an LBA) and an amount of data (such as a number of LBA quanta, e.g., sectors) to write; in response the SSD provides write status and/or requests write data and optionally subsequently provides write status. For yet another example, the commands include a de-allocation command (e.g. a trim command) specifying one or more addresses (such as one or more LBAs) that no longer need be allocated; in response the SSD modifies the Map accordingly and optionally provides de-allocation status. In some contexts an ATA compatible TRIM command is an exemplary de-allocation command. For yet another example, the commands include a super capacitor test command or a data hardening success query; in response, the SSD provides appropriate status. In some embodiments, Host Interfaces  111  is compatible with a SATA protocol and, using NCQ commands, is enabled to have up to 32 pending commands, each with a unique tag represented as a number from 0 to 31. In some embodiments, Tag Tracking  113  is enabled to associate an external tag for a command received via External Interfaces  110  with an internal tag used to track the command during processing by SSD Controller  100 . 
     According to various embodiments, one or more of: Data Processing  121  optionally and/or selectively processes some or all data sent between Buffer  131  and External Interfaces  110 ; and Data Processing  121  optionally and/or selectively processes data stored in Buffer  131 . In some embodiments, Data Processing  121  uses one or more Engines  123  to perform one or more of: formatting; reformatting; transcoding; and any other data processing and/or manipulation task. 
     Buffer  131  stores data sent to/from External Interfaces  110  from/to Device Interfaces  190 . In some embodiments, Buffer  131  additionally stores system data, such as some or all map tables, used by SSD Controller  100  to manage one or more instances of Flash Device  192 . In various embodiments, Buffer  131  has one or more of: Memory  137  used for temporary storage of data; DMA  133  used to control movement of data to and/or from Buffer  131 ; and ECC-X  135  used to provide higher-level error correction and/or redundancy functions; and other data movement and/or manipulation functions. An example of a higher-level redundancy function is a RAID-like capability (e.g. RASIE, such as fractional RASIE and/or non-fractional RASIE, described in further detail elsewhere herein); with redundancy at a flash device level (e.g., multiple ones of Flash Device  192 ) and/or a flash die level (e.g., Flash Die  194 ) instead of at a disk level. 
     According to various embodiments, one or more of: ECC  161  optionally and/or selectively processes some or all data sent between Buffer  131  and Device Interfaces  190 ; and ECC  161  optionally and/or selectively processes data stored in Buffer  131 . In some embodiments, ECC  161  is used to provide lower-level error correction and/or redundancy functions, such as in accordance with one or more ECC techniques. In some embodiments, ECC  161  implements one or more of: a CRC code; a Hamming code; an RS code; a BCH code; an LDPC code; a Viterbi code; a trellis code; a hard-decision code; a soft-decision code; an erasure-based code; any error detecting and/or correcting code; and any combination of the preceding. In some embodiments, ECC  161  includes one or more decoders (such as LDPC decoders). 
     Device Interface Logic  191  controls instances of Flash Device  192  via Device Interfaces  190 . Device Interface Logic  191  is enabled to send data to/from the instances of Flash Device  192  according to a protocol of Flash Device  192 . Device Interface Logic  191  includes Scheduling  193  to selectively sequence control of the instances of Flash Device  192  via Device Interfaces  190 . For example, in some embodiments, Scheduling  193  is enabled to queue operations to the instances of Flash Device  192 , and to selectively send the operations to individual ones of the instances of Flash Device  192  (or Flash Die  194 ) as individual ones of the instances of Flash Device  192  (or Flash Die  194 ) are available. 
     Map  141  converts between data addressing used on External Interfaces  110  and data addressing used on Device Interfaces  190 , using Table  143  to map external data addresses to locations in NVM  199 . For example, in some embodiments, Map  141  converts LBAs used on External Interfaces  110  to block and/or page addresses targeting one or more Flash Die  194 , via mapping provided by Table  143 . For LBAs that have never been written since drive manufacture or de-allocation, the Map points to a default value to return if the LBAs are read. For example, when processing a de-allocation command, the Map is modified so that entries corresponding to the de-allocated LBAs point to one of the default values. In various embodiments, there are various default values, each having a corresponding pointer. The plurality of default values enables reading some de-allocated LBAs (such as in a first range) as one default value, while reading other de-allocated LBAs (such as in a second range) as another default value. The default values, in various embodiments, are defined by flash memory, hardware, firmware, command and/or primitive arguments and/or parameters, programmable registers, or various combinations thereof. 
     In some embodiments, Map  141  uses Table  143  to perform and/or to look up translations between addresses used on External Interfaces  110  and data addressing used on Device Interfaces  190 . According to various embodiments, Table  143  is one or more of: a one-level map; a two-level map; a multi-level map; a map cache; a compressed map; any type of mapping from one address space to another; and any combination of the foregoing. According to various embodiments, Table  143  includes one or more of: static random access memory; dynamic random access memory; NVM (such as flash memory); cache memory; on-chip memory; off-chip memory; and any combination of the foregoing. 
     In some embodiments, Recycler  151  performs garbage collection. For example, in some embodiments, instances of Flash Device  192  contain blocks that must be erased before the blocks are re-writeable. Recycler  151  is enabled to determine which portions of the instances of Flash Device  192  are actively in use (e.g., allocated instead of de-allocated), such as by scanning a map maintained by Map  141 , and to make unused (e.g., de-allocated) portions of the instances of Flash Device  192  available for writing by erasing the unused portions. In further embodiments, Recycler  151  is enabled to move data stored within instances of Flash Device  192  to make larger contiguous portions of the instances of Flash Device  192  available for writing. 
     In some embodiments, instances of Flash Device  192  are selectively and/or dynamically configured, managed, and/or used to have one or more bands for storing data of different types and/or properties. A number, arrangement, size, and type of the bands are dynamically changeable. For example, data from a computing host is written into a hot (active) band, while data from Recycler  151  is written into a cold (less active) band. In some usage scenarios, if the computing host writes a long, sequential stream, then a size of the hot band grows, whereas if the computing host does random writes or few writes, then a size of the cold band grows. 
     CPU  171  controls various portions of SSD Controller  100 . CPU  171  includes CPU Core  172 . CPU Core  172  is, according to various embodiments, one or more single-core or multi-core processors. The individual processors cores in CPU Core  172  are, in some embodiments, multi-threaded. CPU Core  172  includes instruction and/or data caches and/or memories. For example, the instruction memory contains instructions to enable CPU Core  172  to execute programs (e.g. software sometimes called firmware) to control SSD Controller  100 . In some embodiments, some or all of the firmware executed by CPU Core  172  is stored on instances of Flash Device  192  (as illustrated, e.g., as Firmware  106  of NVM  199  in  FIG. 1B ). 
     In various embodiments, CPU  171  further includes: Command Management  173  to track and control commands received via External Interfaces  110  while the commands are in progress; Buffer Management  175  to control allocation and use of Buffer  131 ; Translation Management  177  to control Map  141 ; Coherency Management  179  to control consistency of data addressing and to avoid conflicts such as between external data accesses and recycle data accesses; Device Management  181  to control Device Interface Logic  191 ; Identity Management  182  to control modification and communication of identify information, and optionally other management units. None, any, or all of the management functions performed by CPU  171  are, according to various embodiments, controlled and/or managed by hardware, by software (such as firmware executing on CPU Core  172  or on a host connected via External Interfaces  110 ), or any combination thereof. 
     In some embodiments, CPU  171  is enabled to perform other management tasks, such as one or more of: gathering and/or reporting performance statistics; implementing SMART; controlling power sequencing, controlling and/or monitoring and/or adjusting power consumption; responding to power failures; controlling and/or monitoring and/or adjusting clock rates; and other management tasks. 
     Various embodiments include a computing-host flash memory controller that is similar to SSD Controller  100  and is compatible with operation with various computing hosts, such as via adaptation of Host Interfaces  111  and/or External Interfaces  110 . The various computing hosts include one or any combination of a computer, a workstation computer, a server computer, a storage server, a SAN, a NAS device, a DAS device, a storage appliance, a PC, a laptop computer, a notebook computer, a netbook computer, a tablet device or computer, an ultrabook computer, an electronic reading device (such as an e-reader), a PDA, a navigation system, a (handheld) GPS device, an automotive control system, an automotive media control system or computer, a printer, copier or fax machine or all-in-one device, a POS device, a cash-register, a media player, a television, a media recorder, a DVR, a digital camera, a cellular handset, a cordless telephone handset, and an electronic game. 
     In various embodiments, all or any portions of an SSD controller (or a computing-host flash memory controller) are implemented on a single IC, a single die of a multi-die IC, a plurality of dice of a multi-die IC, or a plurality of ICs. For example, Buffer  131  is implemented on a same die as other elements of SSD Controller  100 . For another example, Buffer  131  is implemented on a different die than other elements of SSD Controller  100 . 
       FIG. 1B  illustrates selected details of various embodiments of systems including one or more instances of the SSD of  FIG. 1A . SSD  101  includes SSD Controller  100  coupled to NVM  199  via Device Interfaces  190 . The figure illustrates various classes of embodiments: a single SSD coupled directly to a host, a plurality of SSDs each respectively coupled directly to a host via respective external interfaces, and one or more SSDs coupled indirectly to a host via various interconnection elements. 
     As an example embodiment of a single SSD coupled directly to a host, one instance of SSD  101  is coupled directly to Host  102  via External Interfaces  110  (e.g. Switch/Fabric/Intermediate Controller  103  is omitted, bypassed, or passed-through). As an example embodiment of a plurality of SSDs each coupled directly to a host via respective external interfaces, each of a plurality of instances of SSD  101  is respectively coupled directly to Host  102  via a respective instance of External Interfaces  110  (e.g. Switch/Fabric/Intermediate Controller  103  is omitted, bypassed, or passed-through). As an example embodiment of one or more SSDs coupled indirectly to a host via various interconnection elements, each of one or more instances of SSD  101  is respectively coupled indirectly to Host  102 . Each indirect coupling is via a respective instance of External Interfaces  110  coupled to Switch/Fabric/Intermediate Controller  103 , and Intermediate Interfaces  104  coupling to Host  102 . 
     Some of the embodiments including Switch/Fabric/Intermediate Controller  103  also include Card Memory  112 C coupled via Memory Interface  180  and accessible by the SSDs. In various embodiments, one or more of the SSDs, the Switch/Fabric/Intermediate Controller, and/or the Card Memory are included on a physically identifiable module, card, or pluggable element (e.g. I/O Card  116 ). In some embodiments, SSD  101  (or variations thereof) corresponds to a SAS drive or a SATA drive that is coupled to an initiator operating as Host  102 . 
     Host  102  is enabled to execute various elements of Host Software  115 , such as various combinations of OS  105 , Driver  107 , Application  109 , and Multi-Device Management Software  114 . Dotted-arrow  107 D is representative of Host Software I/O Device Communication, e.g. data sent/received to/from one or more of the instances of SSD  101  and from/to any one or more of OS  105  via Driver  107 , Driver  107 , and Application  109 , either via Driver  107 , or directly as a VF. 
     OS  105  includes and/or is enabled to operate with drivers (illustrated conceptually by Driver  107 ) for interfacing with the SSD. Various versions of Windows (e.g. 95, 98, ME, NT, XP, 2000, Server, Vista, and 7), various versions of Linux (e.g. Red Hat, Debian, and Ubuntu), and various versions of MacOS (e.g. 8, 9 and X) are examples of OS  105 . In various embodiments, the drivers are standard and/or generic drivers (sometimes termed “shrink-wrapped” or “pre-installed”) operable with a standard interface and/or protocol such as SATA, AHCI, or NVM Express, or are optionally customized and/or vendor specific to enable use of commands specific to SSD  101 . Some drives and/or drivers have pass-through modes to enable application-level programs, such as Application  109  via Optimized NAND Access (sometimes termed ONA) or Direct NAND Access (sometimes termed DNA) techniques, to communicate commands directly to SSD  101 , enabling a customized application to use commands specific to SSD  101  even with a generic driver. ONA techniques include one or more of: use of non-standard modifiers (hints); use of vendor-specific commands; communication of non-standard statistics, such as actual NVM usage according to compressibility; and other techniques. DNA techniques include one or more of: use of non-standard commands or vendor-specific providing unmapped read, write, and/or erase access to the NVM; use of non-standard or vendor-specific commands providing more direct access to the NVM, such as by bypassing formatting of data that the I/O device would otherwise do; and other techniques. Examples of the driver are a driver without ONA or DNA support, an ONA-enabled driver, a DNA-enabled driver, and an ONA/DNA-enabled driver. Further examples of the driver are a vendor-provided, vendor-developed, and/or vendor-enhanced driver, and a client-provided, client-developed, and/or client-enhanced driver. 
     Examples of the application-level programs are an application without ONA or DNA support, an ONA-enabled application, a DNA-enabled application, and an ONA/DNA-enabled application. Dotted-arrow  109 D is representative of Application I/O Device Communication (e.g. bypass via a driver or bypass via a VF for an application), e.g. an ONA-enabled application and an ONA-enabled driver communicating with an SSD, such as without the application using the OS as an intermediary. Dotted-arrow  109 V is representative of Application I/O Device Communication (e.g. bypass via a VF for an application), e.g. a DNA-enabled application and a DNA-enabled driver communicating with an SSD, such as without the application using the OS or the driver as intermediaries. 
     One or more portions of NVM  199  are used, in some embodiments, for firmware storage, e.g. Firmware  106 . The firmware storage includes one or more firmware images (or portions thereof). A firmware image has, for example, one or more images of firmware, executed, e.g., by CPU Core  172  of SSD Controller  100 . A firmware image has, for another example, one or more images of constants, parameter values, and NVM device information, referenced, e.g. by the CPU core during the firmware execution. The images of firmware correspond, e.g., to a current firmware image and zero or more previous (with respect to firmware updates) firmware images. In various embodiments, the firmware provides for generic, standard, ONA, and/or DNA operating modes. In some embodiments, one or more of the firmware operating modes are enabled (e.g. one or more APIs are “unlocked”) via keys or various software techniques, optionally communicated and/or provided by a driver. 
     In some embodiments lacking the Switch/Fabric/Intermediate Controller, the SSD is coupled to the Host directly via External Interfaces  110 . In various embodiments, SSD Controller  100  is coupled to the Host via one or more intermediate levels of other controllers, such as a RAID controller. In some embodiments, SSD  101  (or variations thereof) corresponds to a SAS drive or a SATA drive and Switch/Fabric/Intermediate Controller  103  corresponds to an expander that is in turn coupled to an initiator, or alternatively Switch/Fabric/Intermediate Controller  103  corresponds to a bridge that is indirectly coupled to an initiator via an expander. In some embodiments, Switch/Fabric/Intermediate Controller  103  includes one or more PCIe switches and/or fabrics. 
     In various embodiments, such as some of the embodiments with Host  102  as a computing host (e.g. a computer, a workstation computer, a server computer, a storage server, a SAN, a NAS device, a DAS device, a storage appliance, a PC, a laptop computer, a notebook computer, and/or a netbook computer), the computing host is optionally enabled to communicate (e.g. via optional I/O &amp; Storage Devices/Resources  117  and optional LAN/WAN  119 ) with one or more local and/or remote servers (e.g. optional Servers  118 ). The communication enables, for example, local and/or remote access, management, and/or usage of any one or more of SSD  101  elements. In some embodiments, the communication is wholly or partially via Ethernet. In some embodiments, the communication is wholly or partially via Fibre Channel. LAN/WAN  119  is representative, in various embodiments, of one or more Local and/or Wide Area Networks, such as any one or more of a network in a server farm, a network coupling server farms, a metro-area network, and the Internet. 
     In various embodiments, an SSD controller and/or a computing-host flash memory controller in combination with one or more NVMs are implemented as a non-volatile storage component, such as a USB storage component, a CF storage component, an MMC storage component, an eMMC storage component, a Thunderbolt storage component, a UFS storage component, an SD storage component, a Memory Stick storage component, and an xD-picture card storage component. 
     In various embodiments, all or any portions of an SSD controller (or a computing-host flash memory controller), or functions thereof, are implemented in a host that the controller is to be coupled with (e.g., Host  102  of  FIG. 1B ). In various embodiments, all or any portions of an SSD controller (or a computing-host flash memory controller), or functions thereof, are implemented via hardware (e.g., logic circuitry), software and/or firmware (e.g., driver software and/or SSD control firmware), or any combination thereof. For example, functionality of or associated with an ECC unit (such as similar to ECC  161  and/or ECC-X  135  of  FIG. 1A ) is implemented partially via software on a host and partially via a combination of firmware and hardware in an SSD controller. For another example, functionality of or associated with a recycler unit (such as similar to Recycler  151  of  FIG. 1A ) is implemented partially via software on a host and partially via hardware in a computing-host flash memory controller. 
     Mapping Operation 
       FIG. 2  illustrates selected details of an embodiment of mapping an LPN portion of an LBA. In some embodiments, a read unit is a finest granularity of an NVM that is independently readable, such as a portion of a page of the NVM. In further embodiments, the read unit corresponds to check bits (sometimes-termed redundancy) of a (lower-level) error-correcting code along with all data protected by the check bits. For example, ECC  161  of  FIG. 1A  implements error correction via check bits such as via an LDPC code, and a read unit corresponds to coding bits implementing the LDPC code in addition to data bits protected by the LDPC coding bits. 
     In some embodiments, Map  141  maps LPN  213  portion of LBA  211  to Map Info for LPN  221 , such as via Table  143  (as illustrated in  FIG. 1A ). Map info for an LPN (such as Map Info for LPN  221 ) is sometimes termed a map entry. Map  141  is said to associate an LPN with a corresponding map entry. In various embodiments, mapping is via one or more associative look-ups, via one or more non-associative look-ups, and/or via one or more other techniques. 
     In some embodiments, SSD Controller  100  maintains one map entry for each LPN potentially and/or actively in use. 
     In some embodiments, Map Info for LPN  221  includes respective Read Unit Address  223  and Length in Read Units  225 . In some embodiments, a length and/or a span are stored encoded, such as by storing the length as an offset from the span, e.g. in all or any portions of Length in Read Units  225 . In further embodiments, a first LPN is associated with a first map entry, a second LPN (different from the first LPN, but referring to a logical page of a same size as a logical page referred to by the first LPN) is associated with a second map entry, and the respective length in read units of the first map entry is different from the respective length in read units of the second map entry. 
     In various embodiments, at a same point in time, a first LPN is associated with a first map entry, a second LPN (different from the first LPN) is associated with a second map entry, and the respective read unit address of the first map entry is the same as the respective read unit address of the second map entry. In further embodiments, data associated with the first LPN and data associated with the second LPN are both stored in a same physical page of a same device in NVM  199 . 
     According to various embodiments, Read Unit Address  223  is associated with one or more of: a starting address in the NVM; an ending address in the NVM; an offset of any of the preceding; and any other techniques for identifying a portion of the NVM associated with LPN  213 . 
       FIG. 3  illustrates selected details of an embodiment of accessing an NVM at a read unit address to produce read data organized as various read units, collectively having a length measured in quanta of read units. According to various embodiments, First Read Unit  313  is one or more of: a one of read units in Read Data  311  with a lowest address in an address space of the NVM; a fixed one of the read units; an arbitrary one of the read units; a variable one of the read units; and a one of the read units selected by any other technique. In various embodiments, SSD Controller  100  is enabled to access NVM  199  and produce Read Data  311  by reading no more than a number of read units specified by Length in Read Units  225 . 
       FIG. 4A  illustrates selected details of an embodiment of a read unit (such as Read Units  313  or  315  of  FIG. 3 ) as Read Unit  401 A. In various embodiments and/or usage scenarios, Header 1  411 A through Header N  419 A are contiguous, and respective data regions identified (such as via respective offsets) by each of the headers are contiguous following a last one of the headers. The data regions collectively form Data Bytes  421 A. The data regions are stored in a location order that matches the location order the headers are stored. For example, consider a first header, at the beginning of a read unit, with a second header and a third header contiguously following the first header. A first data region (identified by a first offset in the first header) contiguously follows the third header. A second data region (identified by a second offset in the second header) contiguously follows the first data region. Similarly, a third data region (identified by the third header) contiguously follows the second data region. 
       FIG. 4B  illustrates selected details of another embodiment of a read unit (such as Read Units  313  or  315  of  FIG. 3 ) as Read Unit  401 B. In various embodiments and/or usage scenarios, Header Marker (HM)  410 B is an optional initial field (such as a one-byte field) indicating a number of following contiguous headers (Header 1  411 B, Header 2  412 B . . . Header N  419 B). Data regions (Data Bytes  421 B, Data Bytes  422 B . . . Data Bytes  429 B) are identified respectively by the headers (Header 1  411 B, Header 2  412 B . . . Header N  419 B) and are stored in a location order that is opposite of the location order that the headers are stored. Headers start at the beginning of a read unit, while corresponding data regions start at the end of a read unit. In some embodiments, data bytes within a data region (e.g. Data Bytes  421 B, Data Bytes  422 B . . . Data Bytes  429 B) are arranged in a forward order (byte order matching location order), while in other embodiments, the data bytes are arranged in a reverse order (byte order reversed with respect to location order). In some embodiments, a header marker is used in read units where headers and data bytes are stored in a same location order (e.g. as illustrated in  FIG. 4A ). 
     In some embodiments, Optional Padding Bytes  431 A (or  431 B) are according to granularity of data associated with a particular LPN. For example, in some embodiments, if Data Bytes  421 A (or collectively Data Bytes  421 B, Data Bytes  422 B . . . Data Bytes  429 B) have less than a fixed amount of remaining space, such as 8 bytes, after storing data associated with all but a last one of Header 1  411 A through Header N  419 A (or Header 1  411 B, Header 2  412 B . . . Header N  419 B), then data for an LPN associated with the last header stmls in a subsequent read unit. In further embodiments, a particular offset value (e.g. all ones) in the last header indicates that the data for the LPN associated with the last header starts in the subsequent read unit. 
       FIG. 5  illustrates selected details of an embodiment of a header (such as any of Header 1  411 A through Header N  419 A of  FIG. 4A  or Header 1  411 B through Header  419 B of  FIG. 4B ) having a number of fields. In some embodiments, headers are fixed-length (e.g. each header is a same number of bytes long). Header  501  includes fields Type  511 , Last Indicator  513 , Flags  515 , LPN  517 , Length  519 , and Offset  521 . The type field identifies a category of the data bytes. For example, the type field indicates the category of the data bytes is one of host data (e.g. logical page data) or system data (e.g. map information or checkpoint information). The last field indicates that the header is the last header before the data bytes. In some embodiments with a header marker, the last field is optionally omitted. The LPN field is the LPN that the header is associated with. The LPN field enables parsing of the headers to determine a particular one of the headers that is associated with a particular LPN by, for example, searching the headers for one with an LPN field matching the particular LPN. The length field is the length, in bytes, of the data bytes (e.g. how many bytes of data there are in Data Bytes  421 A associated with Header  501 ). In some embodiments, an offset in the offset field is rounded according to a particular granularity (e.g. 8-byte granularity). 
     In various embodiments, some or all information associated with a particular LPN is stored in a map entry associated with the particular LPN, a header associated with the particular LPN, or both. For example, in some embodiments, some or all of Length  519  is stored in a map entry rather than in a header. 
       FIG. 6  illustrates selected details of an embodiment of blocks, pages, and read units of multiple NVM devices (e.g. one or more flash die and/or flash chips) managed in logical slices and/or sections. The management functions include any one or more of reading, recycling, erasing, programming/writing, and other management functions. The logical slices and/or sections are sometimes referred to as R-blocks. The figure illustrates an embodiment with 66 flash die. Three of the flash die are explicitly illustrated (Flash Die  610 . 65 ,  610 . 1 , and  610 . 0 ) and  63  of the flash die are implicitly illustrated ( 610 . 64  . . .  610 . 2 ). 
     Each of the flash die (such as any one of Flash Die  610 . 65  . . .  610 . 1 , and  610 . 0 ) provides storage organized as blocks (such as Blocks  610 . 65 BB . . .  610 . 65 B 1 , and  610 . 65 B 0  of Flash Die  610 . 65 ; Blocks  610 . 0 BB . . .  610 . 0 B 1 , and  610 . 0 B 0  of Flash Die  610 . 0 ; and so forth). The blocks in turn include pages (such as Pages  610 . 65 PP . . .  610 . 65 P 1 , and  610 . 65 P 0  of Block  610 . 65 B 0 ; Pages  610 . 0 PP . . .  610 . 0 P 1 , and  610 . 0 P 0  of Block  610 . 0 B 0 ; and so forth). The pages in turn include read units (such as Read Units  610 . 65 RR . . .  610 . 65 R 1 , and  610 . 65 R 0  of Page  610 . 65 P 0 ; Read Units  610 . 0 RR . . .  610 . 0 R 1 , and  610 . 0 R 0  of Page  610 . 0 P 0 ; and so forth). 
     In some embodiments, each flash die includes an integer number of blocks (e.g. N blocks) and a block is a smallest quantum of erasing. In some embodiments, each block includes an integer number of pages and a page is a smallest quantum of writing. According to various embodiments, one or more of: a read unit is a smallest quantum of reading and error correction; each page includes an integer number of read units; an associated group of two or more pages includes an integer number of read units; and read units optionally and/or selectively span page boundaries. 
     In various embodiments, various NVM management functions (e.g. reading, recycling, erasing, and/or programming/writing) are performed in units of R-blocks. An R-block is exemplified as a logical slice or section across various die (e.g. all die, all die excluding ones that are wholly or partially failed, and/or one or more selected subsets of die) of, e.g., a flash memory. For example, in a flash memory having R flash die, each flash die having N blocks, each R-block is the block from each of the flash die taken together, for a total of N R-blocks. Continuing with the example, if one of the R flash die fails, then each R-block is the block from each of the flash die except the failed flash die, for a total of N R-blocks, each R-block having one less block than before the failure. For another example, in a flash memory having R flash die, each with N blocks, each R-block is the and (i+1) th  block from each of the flash die, for a total of N/2 R-blocks. For yet another example, in a flash memory having a plurality of dual plane devices, each R-block is the even block and the odd block from each of the dual plane devices. For yet another example, in a flash memory having a plurality of multi-plane devices, each R-block includes blocks selected to maximize parallelism (e.g. during programming) provided by the multi-plane devices. Note that in the aforementioned example of R-blocks in dual plane devices, the dual-plane devices are examples of multi-plane devices, and the R-blocks being the i th  even block and the i th  odd block from each of the dual plane devices is an example of R-blocks including blocks selected to maximize parallelism. For yet another example, in a flash memory having R flash die, an R-block is k non-contiguous blocks, such as blocks i 1 , i 2 , . . . , i k  from each of the R flash die. For a final example, in a flash memory having R flash die, each with N blocks, each R-block is the i th  through (i+k−l) th  block from each of the flash die, for a total of N/k R-blocks. 
     In various embodiments with blocks treated in pairs or other associated groups as part of forming an R-block, respective pages from each block of an associated group of the blocks are also treated as a unit, at least for writing, forming a larger multi-block page. For example, continuing the foregoing dual plane example, a first page of a particular one of the even blocks and a first page of an associated one of the odd blocks are treated as a unit for writing, and optionally and/or selectively as a unit for reading. Similarly, a second page of the particular even block and a second page of the associated odd block are treated as a unit. According to various embodiments, a page of NVM as used herein refers to one or more of: a single page of NVM; a multi-block page of NVM; a multi-block page of NVM for writing that is optionally and/or selectively treated as one or more individual pages for reading; and any other grouping or association of pages of NVM. 
     The figure illustrates a plurality of illustrative R-blocks, three of them explicitly ( 660 . 0 ,  660 . 1 , and  660 .R). Each illustrative R-block is the i th  block from each of the flash die, taken together. E.g., R-block  660 . 0  is Block  610 . 65 B 0  from Flash Die  610 . 65 , block 0 from Flash Die  610 . 64  (not explicitly illustrated), and so forth to Block  610 . 1 B 0  of Flash Die  610 . 1 , and Block  610 . 0 B 0  of Flash Die  610 . 0 . As there are N blocks per flash die, there are thus a total of N R-blocks (R-block  660 .R . . . R-block  660 . 1 , and R-block  660 . 0 ). 
     Another example of an R-block is the i th  block and the (i+l) th  block from each of the flash die, taken together (e.g. Blocks  610 . 65 B 0  and  610 . 65 B 1  from Flash Die  610 . 65 , blocks 0 and 1 from Flash Die  610 . 64  (not explicitly illustrated), and so forth to Blocks  610 . 1 B 0  and  610 . 1 B 1  from Flash Die  610 . 1 , and Blocks  610 . 0 B 0  and  610 . 0 B 1  from Flash Die  610 . 0 ). There are thus N/2 R-blocks, if there are N blocks in each flash die. Yet another example of an R-block is the i th  even and odd blocks from each of a plurality of dual plane devices. Other arrangements of flash die blocks for management as R-blocks are contemplated, including mapping between virtual and physical block addresses to ensure that R-blocks have one block from each die, even if some blocks are inoperable. In various embodiments, some of the N blocks in each flash die are used as spares so that the mapping between virtual and physical block addresses has spare (otherwise unused) blocks to replace defective ones of the blocks in the R-blocks. 
     In various embodiments, reads and/or writes of information in flash die are performed according to an order, such as a ‘read unit first’ order or a ‘page first’ order. An example of a read unit first order for read units illustrated in the figure begins with Read Unit  610 . 0 R 0  followed by  610 . 1 R 0  . . .  610 . 65 R 0 ,  610 . 0 R 1 ,  610 . 1 R 1  . . .  610 . 65 R 1 , and so forth, ending with  610 . 65 RR. An example of a page first order for read units illustrated in the figure begins with Read Unit  610 . 0 R 0  followed by  610 . 0 R 1  . . .  610 . 0 RR,  610 . 1 R 0 ,  610 . 1 R 1  . . .  610 . 1 RR . . .  610 . 65 R 0 ,  610 . 65 R 1 , and so forth, ending with  610 . 65 RR. 
     In various embodiments, a writing and/or a striping order of data within an R-block is page (e.g. lowest to highest) first, across all devices (e.g. lowest to highest numbered devices, as suggested conceptually by Striping Direction  600 ), then the next highest page (across all devices), and so forth, continuing throughout the last page of the R-block. Specifically with respect to R-block  660 . 0 , an example order begins with Page  610 . 0 P 0  (the first page in the first block of Flash Die  610 . 0 ), followed by Page  610 . 1 P 0  (the first page in the first block of Flash Die  610 . 1 ), and so forth continuing to Page  610 . 65 P 0  (the first page in the first block of Flash Die  610 . 65 , and the last block of R-block  660 . 0 ). The example order continues with Page  610 . 0 P 1  (the second page in the first block of Flash Die  610 . 0 ), followed by Page  610 . 1 P 1  (the second page in the first block of Flash Die  610 . 1 ), and so forth continuing to Page  610 . 65 P 1  (the second page in the first block of Flash Die  610 . 65 ). The example continues in an identical order. The example order completes with Page  610 . 0 PP (the last page in the first block of Flash Die  610 . 0 ), followed by Page  610 . 1 PP (the last page in the first block of Flash Die  610 . 1 ), and so forth ending with Page  610 . 65 PP (the last page in the first block of Flash Die  610 . 65 , and the last page in the last block of R-block  660 . 0 ). 
     In various embodiments, Flash Die  610 . 65  . . .  610 . 1 , and  610 . 0  correspond to respective ones of one or more individual Flash Die  194 , of  FIG. 1A . In some embodiments, Flash Die  610 . 65  . . .  610 . 1 , and  610 . 0  are a portion less than all of NVM  199 . For example, in various embodiments, data is striped independently across multiple groups of flash die, and each of the groups of flash die is independently accessible. 
     Higher-Level Redundancy Techniques 
       FIG. 7  illustrates selected details of various embodiments of higher-level redundancy techniques. Flash Device(s)  720  includes 64 flash die (Flash Die  610 . 63 ,  610 . 62 ,  610 . 61  . . .  610 . 0  as explicitly and implicitly illustrated in  FIG. 6 ) and communicates via Interface Channel(s)  730 . Extra Flash Device(s)  740  includes up to two flash die (Flash Die  610 . 65  and  610 . 64  as explicitly and implicitly illustrated in  FIG. 6 ) and communicates via Extra Interface Channel(s)  750 . The Flash Die provide storage for higher-level redundancy information and data storage (e.g. user data and/or user free space) in a storage sub-system, such as NVM in an SSD. (Examples of ‘user data’ in contexts of redundancy information and data storage include all data other than the redundancy information stored on flash memory for later retrieval, such as operating system data, application data, SSD management data, and so forth) Higher-level redundancy enables, e.g., recovering from intermittent or permanent failure of one or more portions of one or more flash die, such as a failure to provide error-corrected data (e.g. via lower-level ECC functions) for a read operation or failure to properly complete a write operation. 
     For example, each flash die (or alternatively each block or each page within each block) is operated in a context of a Redundant Array of Silicon Independent Elements (RASIE). If a failure is detected in a particular flash die (e.g. due to an ECC-uncorrectable read error of a portion of a block of the particular die), then in response, redundant information stored in others of the flash die is used to determine information that would have been provided by the particular die. In some embodiments and/or usage scenarios, sufficient redundant information is stored to enable recovery from one failure within one flash die (during a single operation). Operation in a mode that enables recovery from a single failure includes, in some embodiments, allocating and managing space equivalent to one flash die for higher-level redundancy information, and is termed ‘RASIE-1’. Operation in a mode that enables recovery from two failures includes, in some embodiments, allocating and managing space equivalent to two flash die for higher-level redundancy information, and is termed ‘RASIE-2’. Operation in a mode that enables recovery from three failures includes, in some embodiments, allocating and managing space equivalent to three flash die for higher-level redundancy information, and is termed ‘RASIE-3’. RASIE modes such as RASIE-1, RASIE-2, and RASIE-3 are respective examples of non-fractional RASIE modes, as storage capacity equivalent to an integer multiple of one or more entire flash die (e.g. one, two, and three entire flash die, respectively) is dedicated to higher-level redundancy information. 
     In some embodiments and/or usage scenarios, managing die-level failures is an objective, and spreading information amongst die is performed. For example, higher-level redundancy information is stored in one or more die specifically allocated solely to the higher-level redundancy information. In some embodiments and/or usage scenarios, managing block-level failures is an objective, and spreading information amongst blocks within a die is performed. For example, higher-level redundancy information is stored in one or more blocks allocated to the higher-level redundancy information, the allocation being without regard to which particular die the blocks were part of In some embodiments and/or usage scenarios, managing particular-entity-level failures includes spreading information so that no more than N elements (e.g. one for RASIE-1 and two for RASIE-2) are in any one of the particular entities. Example of the entities include a (packaged) device, a die, an R-block, a block, an R-page (described elsewhere herein), a page, cells associated with a word line, and one or more pluralities of the foregoing. 
     The higher-level redundancy information is computed and written in accordance with (user) data written to the flash die, and is thus available to provide information when a failure is detected. In various embodiments, the higher-level redundancy information is written to the flash die before, after, or in no particular time order with respect to writing of (user) data the higher-level redundancy infomlation is associated with. 
     The figure illustrates various embodiments of non-fractional RASIE operating modes, as summarized in the following table. 
     
       
         
           
               
               
               
             
               
                   
               
               
                 Usage 
                 Usage of Extra 
                 Usage of Flash 
               
               
                 Mode 
                 Flash Device(s) 740 
                 Device(s)720 
               
               
                   
               
             
            
               
                 RASIE-1 
                   
                   
               
            
           
           
               
               
               
               
            
               
                   
                 1-0 
                 -none- 
                 610.63 (1 die)-redundancy 
               
               
                   
                   
                   
                 610.62 . . . 610.0 (63 die)-data storage 
               
               
                   
                 1-1 
                 610.64 (1 die)-redundancy 
                 610.63 . . . 610.0 (64 die)-data storage 
               
            
           
           
               
               
               
            
               
                 RASIE-2 
                   
                   
               
            
           
           
               
               
               
               
            
               
                   
                 2-0 
                 -none- 
                 610.63, 610.62 (2 die)-redundancy 
               
               
                   
                   
                   
                 610.61 . . . 610.0 (62 die)-data storage 
               
               
                   
                 2-1 
                 610.64 (1 die)-redundancy 
                 610.63 (1 die)-redundancy 
               
               
                   
                   
                   
                 610.62 . . . 610.0 (63 die)-data storage 
               
               
                   
                 2-2 
                 610.65, 610.64 
                 610.63 . . . 610.0 (64 die)-data storage 
               
               
                   
                   
                 (2 die)-redundancy 
               
               
                   
               
            
           
         
       
     
     More specifically, in (non-fractional) RASIE-1 modes, space equivalent to one die is allocated to higher-level redundancy information. In RASIE-1 mode 1-0, Extra Flash Device(s)  740  is not used, as higher-level redundancy information is stored in one die of Flash Device(s)  720  (e.g. Flash Die  610 . 63 ), leaving  63  of the die (Flash Die  610 . 62  . . .  610 . 0 ) available for data storage (e.g. user data and/or user free space). In RASIE-1 mode 1-1, one die of Extra Flash Device(s)  740  is used (e.g. Flash Die  610 . 64 ), leaving all of Flash Device(s)  720  (64 die) available for data storage. 
     In (non-fractional) RASIE-2 modes, space equivalent to two die is allocated to higher-level redundancy information. In RASIE-2 mode 2-0, Extra Flash Device(s)  740  is not used, as higher-level redundancy information is stored in two die of Flash Device(s)  720  (e.g. Flash Die  610 . 63  and Flash Die  610 . 62 ), leaving  62  of the die (Flash Die  610 . 61  . . .  610 . 0 ) available for data storage. In RASIE-2 mode 2-1, one die of Extra Flash Device(s)  740  is used (e.g., Flash Die  610 . 64 ), as higher-level redundancy information is partially stored in one die of Flash Device(s)  720  (e.g. Flash Die  610 . 63 ), leaving  63  of the die (Flash Die  610 . 62  . . .  610 . 0 ) available for data storage. In RASIE-2 mode 2-2, two die of Extra Flash Device(s)  740  are used (e.g. Flash Die  610 . 65  and Flash Die  610 . 64 ), leaving all of Flash Device(s)  720  (64 die) available for data storage. 
     In some embodiments, die that are unused in all usage scenarios are unpopulated. For example, in a system operable only in RASIE 2-0 mode and RASIE 1-0 mode (but not in other RASIE modes), Extra Flash Device(s)  740  is unpopulated. 
     In some embodiments, higher-level redundancy information is stored entirely in “dedicated” die (e.g. Flash Die  610 . 63  in RASIE-1 mode 1-0 or Flash Die  610 . 65  and Flash Die  610 . 64  in RASIE-2 mode 2-2). In other embodiments, higher-level redundancy information is stored in any of the die, so for example, in RASIE-1 mode 1-0 Flash Die  610 . 62  is used for higher-level redundancy information, while Flash Die  610 . 63  and Flash Die  610 . 61  . . .  610 . 0  are used for data storage. In some embodiments and/or usage scenarios, higher-level redundancy information is stored in different die (and/or portions thereof) over time, so, for example, in a first time period a first flash die holds higher-level redundancy information while in a second time period a second flash die hold higher-level redundancy information. 
     In various embodiments, there are a plurality of RASIE 1-0 modes (and a plurality of RASIE 2-0 modes) depending on how many flash die are usable. For example, in a first RASIE 1-0 mode (as illustrated in the table above), Flash Die  610 . 63  stores higher-level redundancy information, and Flash Die  610 . 62  . . .  610 . 0  are available for data storage. In a second RASIE 1-0 mode, Flash Die  610 . 63  is no longer usable, Flash Die  610 . 62  stores higher-level redundancy information, and Flash Die  610 . 61  . . .  610 . 0  are available for data storage, decreasing an available amount of data storage by one die. RASIE modes where a die (or any portions thereof) previously available for data storage is no longer available for data storage due to use of the die (or the portions) for higher-level redundancy information are sometimes referred to as reduced-capacity RASIE modes. 
     In some embodiments, higher-level redundancy information is stored using a same and/or a similar lower-level redundancy and/or error correction coding scheme as user data protected by the higher-level redundancy information. Using a lower-level redundancy and/or error correction scheme to protect the higher-level redundancy information enables determining if there is an uncorrectable error in the higher-level redundancy information, in a same and/or a similar manner that an uncorrectable lower-level error in the user data is determined. 
     In some embodiments, higher-level redundancy information is stored in different die for different portions of data. For instance, in some embodiments where flash die are managed in R-blocks, higher-level redundancy information is stored in different flash die for different R-blocks. For example, higher-level redundancy information for an R-block including block 0 of Flash Die  610 . 0  is stored in Flash Die  610 . 0 , while higher-level redundancy information for an R-block including block 1 of Flash Die  610 . 0  is stored in Flash Die  610 . 1 , and so forth. In some embodiments, such as some embodiments where flash die are managed in R-blocks, higher-level redundancy information is written after data the higher-level redundancy information depends on is known and/or is written. 
     In some usage scenarios, one or more portions of an NVM element (e.g. a block of a device, such as Block  610 . 0 BB of Flash Die  610 . 0  of  FIG. 6 ) are, or become during operation, inoperable. In some embodiments, the inoperable portions are mapped out via virtual and physical block addresses (e.g. via processing performed via Map  141  and/or Table  143  of  FIG. 1A .). Alternatively, the inoperable portions are skipped (rather than explicitly mapped out). In some embodiments based on R-blocks, the skipping results in some of the R-blocks having differing numbers of blocks. For example, if Block  610 . 0 BO is defective and unusable, then R-block  660 . 0  has one fewer block than R-block  660 . 1 . The higher-level redundancy information is written in a (per R-block) variable location that is, for example, the last block of each R-block. 
     In various embodiments, one or more elements of  FIG. 7  correspond to one or more elements of  FIG. 1A . For example, Flash Device(s)  720  and Extra Flash Device(s)  740  collectively correspond to NVM  199 , and Interface Channel(s)  730  and Extra Interface Channel(s)  750  collectively correspond to Device Interfaces  190 . For another example, Flash Die  610 . 65  . . .  610 . 0  collectively correspond to the instances of Flash Die  194 . For yet another example, one or more of the flash devices of Flash Device(s)  720  and/or Extra Flash Device(s)  740  correspond to one or more of the instances of Flash Devices  192 . In various embodiments, one or more elements of  FIG. 1A  manage higher-level redundancy information and/or recover user data based at least in part on the higher-level redundancy information in accordance with the RASIE operating modes described herein. For example, a portion of software execution capabilities of CPU  171  is used to manage computation of higher-level redundancy information according to various RASIE operating modes. For another example, Data Processing  121  and/or ECC-X  135  includes hardware elements dedicated to and/or specialized for computation of higher-level redundancy information and/or recovery of user data according to various RASIE operating modes. For yet another example, ECC  161  detects an ECC-uncorrectable (lower-level) read error of a portion of a flash die, and ECC-X  135  detects a RASIE (higher-level) read error and/or enables correction thereof. 
     In various embodiments, Interface Channel(s)  730  variously has one, four, eight, or  16  channels, and Extra Interface Channel(s)  750  variously has one or two channels. In various embodiments, Flash Device(s)  720  is implemented as one, two, four, eight, or  16  devices, each having respectively  64 ,  32 ,  16 , eight, and four of the Flash Die. In various embodiments, Extra Flash Device(s)  740  is implemented as one device having one or two die or as two devices each having one die. In some embodiments, the Flash Die of Extra Flash Device(s)  740  are implemented in devices that also implement the Flash Die of Flash Device(s)  720 . For example, one flash device implements 66 flash die (Flash Die  610 . 65  . . .  610 . 0 ). For another example, two flash devices each implement 33 flash die, e.g. in a first flash device (Flash Die  610 . 65  . . .  610 . 33 ) and in a second flash device (Flash Die  610 . 32  . . .  610 . 0 ). Other arrangements of flash die and flash devices are contemplated. In some embodiments having Extra Flash Device(s)  740  implemented in devices that also implement flash die of Flash Device(s)  720 , the flash die communicate via shared interface channels, or alternatively via interface channels dedicated to particular ones (or sets) of the flash die. While Flash Device(s)  720  and Extra Flash Device(s)  740  are illustrated with specific numbers of flash die (2 and 64, respectively), other embodiments are contemplated, such as Flash Device(s)  720  having 2, 4, 8, 16, 32, or 128 flash die, and/or Extra Flash Device(s)  740  having 0, 1, or 4 flash die. 
       FIG. 8  illustrates selected details of an embodiment of dynamic higher-level redundancy mode management with RASIE, such as dynamically switching between RASIE modes enabled by the various embodiments illustrated by  FIG. 6  and/or  FIG. 7 . In some embodiments and/or usage scenarios, a form of graceful degradation is provided where a storage sub-system (e.g. an SSD) is dynamically transitioned from operating in first higher-level redundancy mode to operating in a second higher-level redundancy mode. The transition is in response to detection of a failure, such as a permanent or intermittent malfunction of an entire flash die or one or more portions thereof, or an operation (such as a read or write operation) thereto. According to various embodiments, the transition is one or more of: global for the SSD; performed on one or more subsets of the SSD; and performed on one or more R-blocks, blocks, and/or pages of the SSD. For example, if a particular block of one of the NVM devices storing RASIE-2 information fails during programming, then subsequent operation of the R-block containing the particular (failed) block transitions to a different higher-level redundancy mode (e.g. a RASIE-1 mode), whereas other R-blocks in the SSD are unaffected and continue to operate in the RASIE-2 mode. 
     With respect to  FIG. 8 , processing begins with higher-level redundancy information and data storage (e.g. user data and/or user free space) arranged in flash die in accordance with a first higher-level redundancy mode (Operate in First Higher-Level Redundancy Mode 802). Flow then proceeds to determine if a failure has been detected (Failure?  803 ), such as a lower-level uncorrectable read error or a write/program failure. If no failure has been detected, then flow proceeds back to continue operation in the first higher-level redundancy mode. If a failure has been detected, then flow proceeds to switch from operating in the first higher-level redundancy mode to operating in a second higher-level redundancy mode (Dynamically Transition Operating Mode 809). 
     The switch begins by (optionally) decreasing space available for data storage (Reduce Free Space  804 ) to account for the failure. If the second higher-level redundancy mode uses sufficiently less higher-level redundancy information than the first higher-level redundancy mode, then the decreasing of available space is omitted. The switch continues by reorganizing data storage in accordance with the second higher-level redundancy mode (Rearrange Data Storage  805 ). The reorganizing includes optionally moving all user data and/or user free space from the flash die where the failure occurred to another one of the flash die (user free space movement is accomplished, in some embodiments, by manipulation of pointers and/or other data structure elements). The switch further continues by selectively restoring (if possible), via the higher-level redundancy information of the first higher-level redundancy mode, any user data that was stored in the flash die where the failure occurred, and writing the restored user data to another one of the flash die, in accordance with the second higher-level redundancy mode (Recover/Store Failed User Data  806 ). The restoring is omitted if the failure is a write/program failure. The switch further continues by optionally computing and writing to the flash die higher-level redundancy information in accordance with the second higher-level redundancy mode (Determine/Store Revised Higher-Level Redundancy Information  807 ). The computing and the writing are omitted if the second higher-level redundancy mode is operable with higher-level redundancy information that was previously in place due to operating in the first higher-level redundancy mode. Then operation begins in the second higher-level redundancy mode (Operate in Second Higher-Level Redundancy Mode 808). 
     The failure detection (Failure?  803 ) is via one or more of: lower-level redundancy and/or error correction (e.g. in accordance with one or more ECC techniques), higher-level redundancy and/or error correction (e.g. in accordance with one or more RASIE techniques), and failing status reported by one or more of the flash die or portions thereof. For example, more than a threshold number of lower-level error corrections of reads within a particular portion (e.g. R-block, block, R-page, page, read unit, or cells associated with a word line) of a particular flash die optionally and/or conditionally results in the particular flash die (or the particular portion) being treated as failed and a higher-level redundancy mode switch is performed so that the failed flash die (or portion) is no longer used. For another example, if a higher-level error correction fails, then an appropriate one of the flash die (or portion thereof) is treated as failed and a higher-level redundancy mode switch is performed so that the failed flash die (or portion) is no longer used. For yet another example, if a flash die returns a program failure status (indicating that a write operation was unsuccessful), then an appropriate block of an appropriate one of the flash die is treated as failed, and optionally and/or conditionally a higher-level redundancy mode switch is performed so that the failed flash die (or alternatively a portion thereof) is no longer used. 
     In some embodiments, a failed block is replaced by remapping via virtual and physical block addresses (e.g. via processing performed via Map  141  and/or Table  143  of  FIG. 1A ). A spare block from a pool of spare blocks is mapped in place of the failed block. Any contents written in the failed block are copied to the replacement block, and writing proceeds in the spare block from where the failure occurred in the failed block. 
     In some embodiments, a failed block is skipped (rather than explicitly remapped), resulting in a “hole” that optionally and/or conditionally results in a higher-level redundancy mode switch when the R-block that the hole is in is next erased (in preparation for re-writing). If the hole is in a location for data storage, then no switch is made, and the hole remains. If the hole is in a location for higher-level redundancy information, then the higher-level redundancy information is stored in another location, and optionally the higher-level redundancy mode is switched. 
     In some embodiments and/or usage scenarios, restoration of user data stored in the flash die where the failure occurred is not possible. For example, if the failure is due to some types of failures detected via higher-level redundancy and/or error correction and/or some types of failing status reported by one or more of the flash die or portions thereof, then some user data is lost. 
     In some embodiments, processing of  FIG. 8  is performed in a context of (e.g. an SSD controller) dynamically transitioning between higher-level redundancy modes in response to a plurality of failures. Specifically, the SSD controller begins operating in a first higher-level redundancy mode and dynamically transitions to a second higher-level redundancy mode in response to a first failure, and subsequently dynamically transitions from the second higher-level redundancy mode to a third higher-level redundancy mode in response to a second failure, and so forth. For instance, an SSD controller operates various flash die in accordance with a RASIE-2 mode 2-2 and dynamically transitions the operation to be in accordance with a RASIE-2 mode 2-1 in response to a first failure. Subsequently, the SSD controller dynamically transitions the operation to be in accordance with a RASIE-2 mode 2-0 in response to a second failure. Further subsequently, the SSD controller dynamically transitions the operation to be in accordance with a RASIE-1 reduced-capacity mode 1-0 in response to a third failure (the reduced-capacity mode 1-0 being similar to RASIE-1 mode 1-0 except with one flash die used for higher-level redundancy information and 62 flash die used for data storage). 
     As a specific example, consider an SSD controller (such as SSD Controller  100  of  FIG. 1A ) coupled to the elements of  FIG. 7 , initially operating in RASIE-2 mode 2-2 (e.g. higher-level redundancy information in Flash Die  610 . 65  and Flash Die  610 . 64 , and data storage in Flash Die  610 . 63  . . .  610 . 0 ), corresponding to operating in the first higher-level redundancy mode. Then a read, or alternatively a write, of one or more of the Flash Die is performed. The read results in an uncorrectable (lower-level) ECC failure, or alternatively the write is unsuccessful, in a portion of a particular one of the Flash Die (e.g. a page of Flash Die  610 . 62  used for user data and/or user free space). In response, the SSD controller dynamically switches from operating in RASIE-2 mode 2-2 to RASIE-2 mode 2-1, no longer using any of Flash Die  610 . 62 . As operation in RASIE-2 mode 2-1 provides 63 die for data storage (versus 64 die in RASIE-2 mode 2-2), space available for data storage is decreased from 64 die to 63 die, and user data and/or user free space is moved accordingly. E.g., all user data from Flash Die  610 . 62  is moved to portions of Flash Die  610 . 63  and Flash Die  610 . 61  . . .  610 . 0  in accordance with user free space. Any user data in the page having the uncorrectable ECC failure is recovered based on the higher-level redundancy information in Flash Die  610 . 65  and/or Flash Die  610 . 64 . Higher-level redundancy information based on data storage in Flash Die  610 . 63  and Flash Die  610 . 61  . . .  610 . 0  and in accordance with RASIE-2 mode 2-1 is computed and stored in Flash Die  610 . 65  and/or Flash Die  610 . 64 . The SSD controller then operates in RASIE-2 mode 2-1 (higher-level redundancy information in Flash Die  610 . 65  and Flash Die  610 . 64 , and data storage in Flash Die  610 . 63  and Flash Die  610 . 61  . . .  610 . 0 ). 
     While the foregoing described several embodiments of dynamic higher-level redundancy mode management with independent silicon elements of a quantum of an entire flash die, other embodiments implement dynamic higher-level redundancy mode management with independent silicon elements that are portions of a die, such as a fraction of an entire flash die (e.g. one-half or one-fourth of a flash die), one or more read units, word line associated cells, pages, R-pages, blocks, or R-blocks. 
     In various embodiments, processing of or relating to one or more elements of  FIG. 8  is performed entirely or in part by one or more elements (or portions thereof) of  FIG. 1A . For example, a portion of software execution capabilities of CPU  171  is used to manage dynamic transitioning between higher-level redundancy modes, such as by directing decreasing space available for data storage or directing reorganizing data storage. For another example, Data Processing  121  and/or ECC-X  135  includes hardware elements dedicated to and/or specialized for computation of higher-level redundancy information in accordance with a ‘target’ redundancy mode. For yet another example, ECC  161  implements lower-level (e.g. ECC) error correction and detection of uncorrectable errors, while ECC-X  135  implements higher-level (e.g. RASIE) error correction and detection of uncorrectable errors and/or memory element failures. For another example, all or any portions of functionality relating to dynamic transitioning between (higher-level) redundancy modes is performed by one or more portions of ECC-X  135 . 
     Higher-Level Redundancy and Adaptive Lower-Level Code Rates 
     In some embodiments and/or usage scenarios, lower-level redundancy and/or error correction uses an adaptive code rate (e.g. an adaptive ECC technique using a variable code rate). For instance, a first read unit is managed with a first code rate that provides relatively more usable data bits than a second read unit that is managed with a second code rate. In some embodiments and/or usage scenarios with lower-level redundancy and/or error correction using a variable code rate, higher-level redundancy information is stored in portions of independent silicon elements (such as portions of flash die) that are managed with lower-level redundancy and/or error correction that provides relatively more usable data bits or the most usable data bits with respect to data storage protected by the higher-level redundancy information. The portion(s) where the higher-level redundancy information is stored varies, in various embodiments and/or usage scenarios, on a per R-block basis, on a per-die basis, dynamically over time, or any combination thereof. In various embodiments, one or more die, R-blocks, blocks, and/or pages are selected for storage of higher-level redundancy data based on having the most usable data with respect to lower-level error correction code rate. 
     For example, higher-level redundancy information is stored at varying locations (such as block locations) on a per R-block basis, the locations (such as blocks) being those having the most usable data based on lower-level error correction code rate. In an arbitrary example, consider a RASIE-2 operating mode scenario in the context of  FIG. 7 , wherein the collection of 66 flash die is treated as a logical “stack” of up to M R-blocks of one-block height each, where M is the number of blocks per flash die. (In the most conceptually straightforward case, each R-block in the stack is made up of the same physical block number from each die, but to accommodate failed blocks this constraint is lifted in at least some embodiments. In yet other embodiments, the constraint is maintained, but “holes” are accommodated corresponding to the failed blocks.) Thus, each R-block has up to 66 blocks, corresponding to one block from each of Flash Die  610 . 0  through Flash Die  610 . 65 . (In  FIG. 7 , while only some flash die in the range  610 . 0  through  610 . 65  are explicitly enumerated, it is understood by the use of ellipsis that all flash die in this range are implicitly illustrated.) The higher-level redundancy information is written into whichever blocks of each R-block have the most useable data based on lower-level error correction code rate. If for example in a first R-block, blocks corresponding to Flash Die  610 . 15  and  610 . 49  happen to have the most usable data based on lower-level error correction code rate, then higher-level redundancy information is written into those blocks (corresponding to Flash Die  610 . 15  and  610 . 49 ). While if in a second R-block, blocks corresponding to Flash Die  610 . 9  and  610 . 35  happen to have the most usable data based on lower-level error correction code rate, then higher-level redundancy information is written into those blocks (corresponding to Flash Die  610 . 9  and  610 . 35 ). In some embodiments, the higher-level redundancy data is written after all other data in an R-block is known. 
       FIG. 9  illustrates an embodiment of Read Units ( 911 ,  931  . . .  951 ,  971 ) having lower-level redundancy information of adaptive (e.g. differing and/or varying over time and/or by location) code rates protected by higher-level redundancy information stored in one or more of the read units. Each of the read units has a portion enabled and/or allocated to contain User Data ( 911 .U,  931 . 0  . . .  951 .U,  971 .U), and a remaining portion enabled and/or allocated to contain lower-level redundancy information, such as check bits of an ECC code as Lower-Level ECC ( 911 .E,  931 .E . . .  951 .E,  971 .E). 
     In the figure, vertical dimensions are relatively to scale and indicative of relative size. Thus Read Units  911  and  951  are of a same first size (in some embodiments, all read units are a same size for all blocks of all NVM devices), Lower-LevelECC portions  931 .E and  951 .E are of a same second size, and Lower-Level ECC portions  911 .E and  971 .E are of a same third size. Read Unit  931  is larger than Read Units  911  and  951  that are in turn larger than Read Unit  971 . User Data portion  931 . 0  is larger than User Data portion  951 .U. User Data portion  951 . 0  is larger than User Data portion  911 .U. Lower-Level ECC portion  951 .E is smaller than Lower-Level ECC portion  911 .E. 
     As illustrated in the figure, respective read units have respective sizes, e.g. per respective block of NVM, enabling varying lower-level code rates as used in the respective blocks. More specifically, Read Units  931  and  951  have a same amount of lower-level redundancy information ( 931 .E and  951 .E are a same size), but a lower-level code rate of Read Unit  931  is higher than a lower-level code rate of Read Unit  951 , as Read Unit  931  contains more User Data ( 931 .U) than Read Unit  951  (containing User Data  951 .U). 
     As illustrated in the figure, respective read units have respective sizes of user data, enabling various sizes of user data in each of two or more read units of a same size. For example, a size of user data is varied to change a lower-level code rate used in a particular read unit. More specifically, Read Units  951  and  911  have a same size, but have different respective amounts of User Data ( 951 . 0  and  911 .U), and thus different respective amounts of lower-level redundancy information ( 951 .E and  911 .E), enabling Read Unit  951  to have a higher lower-level code rate than Read Unit  911 . 
     In some embodiments and/or usage scenarios, varying and/or changing a lower-level code rate advantageously enables providing a sufficient amount of lower-level ECC information to achieve lower-level error correction requirements while maximizing an amount of user data. 
     In some embodiments with a varying amount of user data in read units, higher-level redundancy information is stored in a one or more of the read units having a largest amount of user data. For example in  FIG. 9 , using a RASIE-1 mode, higher-level redundancy information is stored in User Data  931 .U, and using a RASIE-2 mode, higher-level redundancy information is stored in User Data  931 . 0  and User Data  951 .U. Storing the higher-level redundancy information in read units with the largest amount of user data (among the read units protected by the higher-level redundancy information) ensures that there is sufficient higher-level redundancy information to protect the user data in all of the other read units. 
     According to various embodiments, one or more techniques are used to determine which of one or more read units among a number of read units protected by higher-level redundancy information are used to store higher-level redundancy information. In a first example, the latest-written one (for RASIE-1) or two (for RASIE-2) read units that have a largest amount of user data are used. In a second example, the earliest-written one (for RASIE-1) or two (for RASIE-2) read units that have a largest amount of user data are used. Similarly, any technique to deterministically select one or more read units having a largest amount of user data so as to protect all of the remaining user data in the other read units is within the scope of the techniques considered herein. 
     Higher-Level Redundancy Information Computation Techniques 
     In various embodiments and/or usage scenarios, higher-level redundancy information is computed with a variety of techniques, such as via parity, RS, and/or weighted-sum techniques. For example, in some higher-level redundancy modes enabling recovery from one (lower-level) failure (e.g. RASIE-1), higher-level redundancy information is computed via parity techniques. For another example, in some higher-level redundancy modes enabling recovery from two (lower-level) failures (e.g. RASIE-2), higher-level redundancy information is computed via a combination of parity and RS techniques. A first portion of the higher-level redundancy information is computed using parity coding and a second portion is computing using RS coding. For yet another example, in some higher-level redundancy modes enabling recovery from two failures (e.g. RASIE-2), higher-level redundancy information is computed via a combination of parity and weighted-sum techniques. A first portion of the higher-level redundancy information is computed using parity coding and a second portion is computing using weighted-sum coding. The higher-level redundancy information is managed, e.g. via reads and writes of pages of NVM, using lower-level failure detection techniques (such as ECC) identical to or similar to lower-level failure detection techniques used for pages of the NVM available for storing data protected by the higher-level redundancy information. 
     As a specific example for RASIE-2, a first page of higher-level redundancy information is computed using parity coding via an XOR of all first pages in a stripe across an R-block. More specifically, an XOR is performed of all of the first bytes of all of the first pages in the stripe across the R-block, resulting in a first byte of the first page of higher-level redundancy information. Similarly, a second byte of higher-level redundancy information is formed by XORing all of the second bytes of all of the first pages in the stripe across the R-block, and so forth for all of the bytes of all of the first pages in the stripe. A second page of higher-level redundancy information is computed using a weighted-sum technique as follows. 
     Arithmetic is performed over a finite field, such as a Galois Field (used as an example). Examples assume data being operated on is byte-wide, and thus a field such as GF(256) is used. In various embodiments, data is operated on in any units. 
     Each page in a stripe is assigned a unique non-zero “index”. The values of the indices are chosen to simplify implementation complexity, and are not related to any form of generator polynomial. For example, the pages are labeled (e.g. by software) by die location in a stripe from 0 to N−1, and a suitable value for the indices is the ones-complement of the die number (ensured to be non-zero provided N&lt;255). Another selection of index values is the lowest-weight (fewest number of set bits or fewest number of clear bits) non-zero integers, e.g. to reduce and/or minimize hardware costs. Selecting gray-coded values for the indices, in some embodiments and/or usage scenarios, minimizes transitions and/or reduces power as pages are processed. 
     The index values are not selected according to finite field arithmetic, but are selected according to other principles. Notwithstanding this, each index value corresponds to a non-zero element in the finite field. Assume that page i has index value Ki (and page j has index value Kj). The weighted-sum redundancy is the (GF field) sum (over corresponding bytes Pi from each page i) of Ki*Pi, each byte multiplied (over the GF field) by its index value. 
     Thus weighted-sum redundancy information is computed for each byte as: 
     R0=sum over all corresponding bytes Pi; 
     R1=sum over all corresponding bytes Ki*Pi; 
     R0 is the XOR of all the corresponding bytes; and 
     R1 is a weighted sum of the bytes, where the weights are selected as the index values. The foregoing computation is iterated for each of the corresponding bytes in a page, producing corresponding pages of bytes for each of R0 and R1. In the following discussion, R0 and R1 are described in some contexts respectively as single elements (e.g. each being a single byte) for clarity of exposition, but as in the foregoing computation, each are representative of a respective page of elements (e.g. each being a page of bytes). 
     Pi represents a byte in page i, and Pj represents a byte in page j. Processing is described with respect to one stripe of corresponding bytes from each page, and iteration is performed over all corresponding bytes. If some pages are “shorter” than others, due to for example to having a different (lower-level) redundancy code rate, then the shorter pages are zero-padded (or padded by any known value used the same way on encode and decode) so that all pages that are processed effectively have the same size. 
     Summations to compute R0 and R1 are performable in any order, via various serial and/or parallel computations, according to various embodiments. Pages do not have to be processed in any specific order as whether Ki*Pi is added in prior to or subsequent to Kj*Pj has no effect on the result in R1. Computation of R0 and R1 values corresponding to various bytes of a page of redundancy information are independent of each other and are computable in any order, via various serial and/or parallel computations, according to various embodiments. Fmlher, subtracting Ki*Pi from R1 (and subtracting Pi from R0) enables “backing out” of computation effects on pages. Since over a GF field, addition and subtraction are both XOR (thus subtracting is equivalent to simply adding in a second time), in some embodiments and/or usage scenarios, no special hardware is needed for GF field implementations (e.g. a logical XOR capability is sufficient) to “back out” a page. 
     In the event of an uncorrectable lower-level error, higher-level error correction begins, in some embodiments, by re-computing R0 and R1, but by omitting page(s) (sometimes referred to as column(s)) that have uncorrectable lower-level errors. Correction proceeds by subtracting the recomputed R0 from the original R0 to produce A R0, and subtracting the recomputed R1 from the original R1 to produce A R1. 
     If there are no uncorrectable lower-level errors, then the recomputed R0 and R1 are both zero. If there are uncorrectable lower-level errors, then the recomputed R0 and R1 (after the subtraction) reflect the “missing” data (that was not added in the second time, but was present in the original values). 
     If there is one uncorrectable lower-level error, then the recomputed R0 is used to correct the error (and the recomputedR1 is not needed). 
     If there are two uncorrectable lower-level errors, then the recomputed R0 and R1 are used to correct the errors. If both the pages of R0 and R1 values are the pages with uncorrectable lower-level errors, then no correction of data storage pages is needed. If the page of R1 values is one of the pages with uncorrectable lower-level errors, then correction is via R0 (the recomputed R0 value is the value of the data storage page with uncorrectable lower-level errors). 
     If there are two uncorrectable lower-level errors in data storage pages, or if the RO page is one of the pages with uncorrectable lower-level errors, then correction starts by computing Δ RO and Δ R1 as above. If the R0 page is one of the pages with uncorrectable lower-level errors, computation of the 8RO page is optionally omitted. If page i and page j are pages with uncorrectable lower-level errors, then the recomputed Δ RO=Pi+Pj, and the recomputed Δ R1=Ki*Pi+Kj*Pj. Equation solving produces:
 
 Pi =(Δ R 1− Kj*ΔR 0)/( Ki−Kj )
 
 Pj=ΔRO−Pi  
 
If R0 is one of the pages with uncorrectable lower-level errors, then (because R0 is not included in R1), R1=Ki*Pi, or Pi=R1/Ki; the same result obtained by setting Kj=0 in the formulas above (to ignore R0).
 
     In an alternative embodiment, a finite field defined by integers mod p, where p is a prime, is used instead of a Galois Field. The computations are identical to those described above, with addition being integer addition mod p, and multiplication being integer multiplication mod p. For example, if pages are pages of bytes, a finite field of integers mod 257 is used. All user data bytes are in a range 0-255 and are stored in one byte each. R1 results, however, have values ranging from 0-256, requiring more than one byte for representation. There are many ways to encode values from 0-256 to minimize storage space and enable the R1 page to be stored with reduced overhead. For example, values 0 and 256 are stored as nine-bit sequences 000000000 and 000000001 respectively, and all other values are stored in eight bits. Given a random distribution of R1 values, storage overhead is &lt;0.1%. As described with respect to  FIG. 9 , the R1 page is selected to have a largest amount of user data, enabling the storage overhead to be hidden in some usage scenarios. 
       FIG. 10  illustrates selected details of an embodiment of higher-level redundancy information result and data source correspondences, for instance as used by RASIE-2 mode 2-2 in a context such as  FIG. 7  and as further illustrated in  FIG. 6 .  FIG. 10  explicitly illustrates Flash Die  610 . 0 ,  610 . 63 ,  610 . 64  and,  610 . 65 , and by ellipsis ( . . . ) implicitly illustrates Flash Die  610 . 1  . . .  610 . 62 . Selected details of blocks, pages, and bytes within pages of the Flash Die are illustrated. A first portion of higher-level redundancy information is illustrated as R0  1010  (stored in Flash Die  610 . 64 ), and is computed using parity coding. A second portion of higher-level redundancy information is illustrated as R1  1011  (stored in Flash Die  610 . 65 ), and is computed using weighted-sum coding. Storage for data information is illustrated as Data  1019  (stored in Flash Die  610 . 0  . . .  610 . 63 ). 
     Dashed-arrow  1001  conceptually indicates a two-byte redundancy computation result (one byte for each of R0  1010  and R1  1011 ) based on corresponding first bytes of all first pages (across all flash die) of Data  1019 . As illustrated, the two-byte result is the first byte on each of the first R0 and R1 pages. Dashed arrow  1002  conceptually indicates a two-byte redundancy computation result (one byte for each of R0  1010  and R1  1011 ) based on corresponding last bytes of all first pages of Data  1019 . As illustrated, the two-byte result is the last byte on each of the first R0 and R1 pages. Dashed-arrow  1003  conceptually indicates a two-page redundancy computation result (one page for each of R0  1010  and R1  1011 ) based on corresponding last pages of the first blocks (across all flash die) of Data  1019 . 
     Note that as described elsewhere herein, in some embodiments higher-level redundancy information is stored in different die for different portions of data. Thus, R0 and R1, in some embodiments, are stored across various die, rather than two “dedicated” die, as illustrated in  FIG. 10 . 
       FIG. 11  illustrates selected details of an embodiment of higher-level redundancy information computations, for instance as used by RASIE-2 mode 2-2 in a context such as  FIG. 7  and as further illustrated in  FIG. 6  and  FIG. 10 , with various operating conditions according to various embodiments. More specifically,  FIG. 11  illustrates parity coding computation for a byte of R0 and weighted-sum coding computation for a byte of R1, in accordance with, e.g., the two-byte redundancy computation result illustrated conceptually by dashed-arrow  1001  of  FIG. 10 . The operating conditions include one or more of: performing arithmetic over a finite field (such as a Galois Field), indices (corresponding to dummy summation variable “i” values in  FIG. 11 ) being ones-complement of flash die number (or any other unique and non-zero numbering), and indices corresponding to non-zero elements in the finite field. The computation as illustrated in  FIG. 11  is repeated for all of the bytes of R0 and R1, based on the corresponding data bytes. Note that non-zero indices enable R1 values to include a contribution from every element of Pi. 
     Thus there are no dependencies on computing any of the bytes of R0 on each other or between any of the bytes of R1. Therefore various embodiments are contemplated where R0 and R1 values are computed entirely (massively) in parallel, partially in parallel, or entirely in serial. For example, all of the R0 and/or R1 bytes of a page are computed in parallel. For another example, all of the R0 bytes of a page are computed in parallel followed (or preceded) by computing in parallel all of the R1 bytes of a page. 
     Further, there are no ordering requirements on computing any of the bytes of R0 with respect to each other or with respect to computing any of the bytes of R1. Therefore various embodiments are contemplated where R0 and R1 values are computed entirely in-order with respect to each other, in-order with respect to each other separately (e.g. R0 computations are in-order with respect to each other but with no ordering with respect to R1, and vice-versa), or with no particular ordering (e.g. entirely out-of-order, unordered, or randomly ordered). For example, all of the R0 bytes of a page are computed in a particular order (e.g. from lowest byte to highest byte), followed by all of the R1 bytes of the page in the particular order. For another example, all of the R0 bytes of a page are computed in the particular order, and independently all of the R1 bytes of the page are computed in the particular order. For yet another example, all of the R0 bytes of a page as well as all of the R1 bytes of a page are computed in no particular order with respect to each other (e.g. as data operands become available). 
     For yet another example, all of the R0 and R1 bytes of one or more pages are computed in an order determined by an order of completion of one or more read operations performed on one or more NVMs (each having, e.g., one or more flash die), the read operations for reading the data bytes referenced by the summation and weighted-summation computations (Pi) illustrated in  FIG. 11 . Performing the computations in the order determined by the completion of the read operations enables, in some embodiments and/or usage scenarios, reduction or elimination of buffering between the NVMs and elements enabled to perform the computations. Performing the computations in the order determined by the completion of the read operations enables, in some embodiments and/or usage scenarios, reduction in memory bandwidth used to perform the computations, such as memory bandwidth of the NVMs. 
     For yet another example, all of the R0 and R1 bytes of one or more pages are computed in an order determined by an order of data returned and/or data available in response to read operations performed on one or more NVMs (each having, e.g., one or more flash die), the data returned and/or data available corresponding to the data bytes referenced by the summation and weighted-summation computations (Pi) illustrated in  FIG. 11 . In some embodiments and/or usage scenarios, performing the computations in the order determined by the data returned and/or data available reduces or eliminates buffering between the NVMs and elements enabled to perform the computations. In some embodiments and/or usage scenarios, performing the computations in the order determined by the data returned and/or data available reduces memory bandwidth used to perform the computations, such as memory bandwidth of the NV Ms. In some embodiments, the read operations are performed in a particular order (e.g. from lowest byte to highest byte of Pi), while in other embodiments, the read operations are performed in no particular order. 
       FIG. 12  illustrates selected details of an embodiment of recovery from one (lower-level) failure (during a single operation), for instance in a context such as associated with  FIG. 11 , and where the lower-level failure has occurred on page j. Note that if the lower-level failure is on an R0 or an R1 page, then R0 (or R1) is re-determined as described by  FIG. 11 .  FIG. 12  illustrates a computation for one byte of a recovered value for page j (note that the summation omits page j where the failure occurred). The computation as illustrated in  FIG. 12  is repeated to determine recovered values for all of the bytes of page j, based on corresponding bytes of R0 and corresponding bytes from all of the data pages except for page j. 
     Thus there are no dependencies on computing any of the bytes of the recovered values of page j on each other. Therefore various embodiments are contemplated where Pj recovery values are computed ranging from highly parallel to highly serial, similarly to the aforementioned computations for R0 and R1 values. Further, there are no order requirements on computing any of the recovery values of page j with respect to each other. Therefore various embodiments are contemplated where Pj recovery values are computed in varying orderings with respect to each other, similarly to the aforementioned computations for R0 and R1 values. 
     Some exemplary embodiments perform computations related to determining recovery values in orderings at least in part determined by an ordering of completion of one or more read operations performed on one or more NVMs (each having, e.g., one or more flash die), the read operations for reading the NVMs to obtain R0 and/or Pi values as illustrated by  FIG. 12 . Performing the computations in the order of the completion of the read operations enables, in some embodiments and/or usage scenarios, reduction or elimination of buffering between the NVMs and elements enabled to perform the computations. Performing the computations in the order of the completion of the read operations enables, in some embodiments and/or usage scenarios, reduction in memory bandwidth used to perform the computations, such as memory bandwidth of the NVMs. 
     Some exemplary embodiments perform computations related to determining recovery values in orderings at least in part determined by an ordering of data returned and/or data available from one or more NVMs (each having, e.g., one or more flash die), the data returned and/or data available being in response to read operations performed on the NVMs to obtain R0 and/or Pi values as illustrated by  FIG. 12 . In some embodiments and/or usage scenarios, performing the computations in the order of the data returned and/or data available from the read operations reduces or eliminates buffering between the NVMs and elements enabled to perform the computations. In some embodiments and/or usage scenarios, performing the computations in the order of the data returned and/or data available from the read operations reduces memory bandwidth used to perform the computations, such as memory bandwidth of the NVMs. In some embodiments, the read operations are performed in a particular order (e.g. from lowest byte to highest byte of Pi), while in other embodiments, the read operations are performed in no particular order. 
       FIGS. 13A-13D  illustrate selected details of an embodiment of recovery from two (lower-level) failures (during a single operation), for instance in a context such as associated with  FIG. 11 , and where the lower-level failures have occurred on pages m and n. Note that if the lower-level failures are on R0 and R1 pages, then R0 and R1 are unneeded for data recovery, and there is no processing to recover data. Otherwise, note that if one of the lower-level failures is an R1 page, then data recovery proceeds as described by  FIG. 12 . If neither of the lower-level failures are on R0 or R1 pages, then recovery of Pm and Pn values proceeds as follows.  FIG. 13A  illustrates computations for one byte of a revised R0 as R0′ and for one byte of a revised R1 as R1′ (note that the summations omit pages m and n where the failures occurred).  FIG. 13B  illustrates equalities relating one byte of the original redundancy information (R0, R1) to the revised redundancy information (R0′, R1′), and the terms omitted from the summations used to form the revised R0 and R1 (Pm+Pn and Km*Pm+Kn*Pn).  FIG. 13C  illustrates an algebraic rearrangement of  FIG. 13B , with introduced terms delta R0 (ΔR0) and delta R1 (ΔR1).  FIG. 13D  illustrates a solution of  FIG. 13C  for Pn and Pm, and thus represents computations for one byte of a recovered value for page n and one byte of a recovered value for page m, based on the corresponding bytes of R1, R0, and corresponding bytes from all of the data pages except for pages m and n. Note that unique indices enable a non-zero denominator, as Kn is unique with respect to Km. 
     In various embodiments, computations as represented by  FIGS. 13A-13D  are performed to determine one byte of a recovered value for page m and one byte of a recovered value for page n. The computations are repeated to determine recovered values for all of the bytes of pages m and n, based on corresponding bytes of RO, R1, and corresponding bytes from all of the data pages except for pages m and n. 
     Thus there are no dependencies on computing any of the bytes of the recovered values of pages m or n on each other. Therefore various embodiments are contemplated where Pm and/or Pn recovery values are computed ranging from highly parallel to highly serial, similarly to the aforementioned computations for Pj recovery values. Further, there are no order requirements on computing any of the recovery values of page m and/or page n with respect to each other. Therefore various embodiments are contemplated where Pm and/or Pn recovery values are computed in varying orderings with respect to each other, similarly to the aforementioned computations for Pj recovery values. 
     Some exemplary embodiments perform computations related to determining recovery values (such as computations for R0′ and R1′) in orderings at least in part determined by an ordering of completion of one or more read operations performed on one or more NVMs (each having, e.g., one or more flash die), the read operations to obtain any one or more of values illustrated as sources for computations in any of  FIGS. 13A-13D . Performing the computations in the order of the completion of the read operations enables, in some embodiments and/or usage scenarios, reduction or elimination of buffering between the NVMs and elements enabled to perform the computations. Performing the computations in the order of the completion of the read operations enables, in some embodiments and/or usage scenarios, reduction in memory bandwidth used to perform the computations, such as memory bandwidth of the NVMs. 
     Some other exemplary embodiments perform computations related to determining recovery values (such as computations for R0′ and R1′) in orderings at least in part determined by an ordering of data returned and/or data available from one or more NVMs (each having, e.g., one or more flash die), the data returned and/or data available being in response to read operations performed on the NVMs to obtain any one or more of values illustrated as sources for computations in any of  FIGS. 13A-13D . In some embodiments and/or usage scenarios, performing the computations in the order of the data returned and/or data available from the read operations reduces or eliminates buffering between the NVMs and elements enabled to perform the computations. In some embodiments and/or usage scenarios, performing the computations in the order of the data returned and/or data available from the read operations reduces memory bandwidth used to perform the computations, such as memory bandwidth of the NVMs. In some embodiments, the read operations are performed in a particular order (e.g. from lowest byte to highest byte of Pi), while in other embodiments, the read operations are performed in no particular order. 
       FIGS. 14A and 14B  illustrate selected details of an embodiment of computing higher-level redundancy information with respect to pages received from NVMs.  FIG. 14A  illustrates selected details of an embodiment of sending a plurality of read commands to one or more NVMs via issuing (Issue Read Operation  1402 A), checking if all of the commands have been sent (All Issued?  1403 A), and if not, then looping back to send another of the commands. Note that other embodiments are contemplated where a plurality of commands are issued at a time, rather than one at a time. 
       FIG. 14B  illustrates selected details of an embodiment of processing pages received from the NVMs in response to the read commands sent as illustrated in  FIG. 14A . A check is made to determine if a page is available (Page Ready?  1402 B). If not, then processing loops back to perform the check again. If a page is available, then higher-level redundancy information processing relating to the page is carried out (Perform Page-Based Computations  1403 B). Then a check is made to determine if all pages have been processed (Pages Finished?  1404 B). If so, then processing is complete (End  1499 B), otherwise flow loops back to determine if another page is available. 
     Other than reception of pages in response to the commands sent as illustrated in  FIG. 14A , the processing illustrated in  FIG. 14B  is independent of processing illustrated in  FIG. 14A . In various scenarios, arrival order of the pages varies according to the NVM type, state, operating environment, and other factors, and in some circumstances is different than sending order or arrival order of the read commands corresponding to the arriving pages. As processing of  FIG. 14A  is independent of  FIG. 14B  (other than page data arrival being dependent on a corresponding read command being sent), in some embodiments and/or usage scenarios, read commands are being sent ( FIG. 14A ) while read data is being received/processed ( FIG. 14B ). In some embodiments and/or usage scenarios, some of the pages are provided from one or more buffers rather than being requested via read commands directed to the NVMs, for example if a particular page is present in a particular buffer before a read command for the particular page is to be sent. In some embodiments and/or usage scenarios, pages other than in response to the commands sent are provided by the NVMs intermixed with the pages provided in response to the commands sent, e.g. pages provided in response to read commands sent for other activities. 
     In various embodiments, computations for R0 and R1, as illustrated by  FIG. 11 , are performed at least in part as illustrated by  FIGS. 14A and 14B . For a first example, read commands for all data pages necessary to compute corresponding R0 and R1 pages are sent to one or more NVMs as illustrated by  FIG. 14A . The pages of data received in response to the read commands are processed as the pages are received to compute the R0 and the R1 pages, as illustrated by  FIG. 11 . For a second example, read commands for a pair (corresponding, e.g., to two planes of a dual-plane NVM) of R0 and R1 pages are sent to one or more NVMs as illustrated by  FIG. 14A . The pages of data received in response to the read commands are processed as the pages are received to compute the R0 and the R1 pages, as illustrated by  FIG. 11 . 
     In various embodiments, computations for Pj, as illustrated by  FIG. 12 , are performed at least in part as illustrated by  FIGS. 14A and 14B . For a first example, read commands for all data pages necessary to compute a particular Pj page are sent to one or more NVMs as illustrated by  FIG. 14A . The pages of data received in response to the read commands are processed as the pages are received to compute the Pj page, as illustrated by  FIG. 12 . For a second example, read commands for a pair of Pj pages (corresponding, e.g., to two planes of a dual-plane NVM) are sent to one or more NVMs as illustrated by  FIG. 14A , and the pages of data received are processed as received, as illustrated by  FIG. 14B , to compute the pair of Pj pages. 
     In various embodiments, computations related to determining recovery values (such as computations for R0′ and R1′), as illustrated by any of  FIGS. 13A-13D , are performed at least in part as illustrated by  FIGS. 14A and 14B . For a first example, read commands for all data pages necessary to compute a particular R0′ page and a particular R1′ are sent to one or more NVMs as illustrated by  FIG. 14A . The pages of data received in response to the read commands are processed as the pages are received to compute the R0′ and R1′ pages, as illustrated by  FIG. 13A . For a second example, read commands for a pair of R0′ and R1′ pages (corresponding, e.g., to two planes of a dual-plane NVM) are sent to one or more NVMs as illustrated by  FIG. 14A , and the pages of data received are processed as received, as illustrated by  FIG. 14B , to compute the pair of R0′ and R1′ pages. 
       FIGS. 15A-15C  illustrate selected details of an embodiment of backing out of a computation of higher-level redundancy information with respect to a write provided to NVMs, for instance in a context such as associated with  FIG. 11 .  FIG. 15A  illustrates selected details of an embodiment of sending a plurality of write commands to one or more NVMs via issuing (Issue Write Operation  1502 A), checking if all of the commands have been sent (All Issued?  1503 A), and if not, then looping back to send another of the commands. Note that other embodiments are contemplated where a plurality of commands are issued at a time, rather than one at a time. 
       FIG. 15B  illustrates selected details of an embodiment of processing write completion and status information received from the NVMs in response to the write commands sent as illustrated in  FIG. 15A . A check is made to determine if a write has completed without errors (Write OK?  1502 B). If so, then a check is made to determine if all writes have been completed (Writes Finished?  1504 B). If so, then processing is complete (End  1599 B). If a write has been completed but with a (lower-level) error such as a program failure, then the flow proceeds to “undo” the effect of the write with respect to higher-level redundancy information computation (Backout Write from Redundancy  1503 B). More specifically, data for the write with the lower-level error is de-computed from any corresponding higher-level redundancy information computations (assuming that the data for the write had already been included in the corresponding higher-level redundancy computations under a presumption that the write would succeed). For example, a lower-level write failure is detected on a particular page j. In response, revised R0 and R1 pages are computed such that page j data (Pj) is set to zero.  FIG. 15C  illustrates selected details of an embodiment of a computation for a single byte of a revised R0 (nR0) and a single byte of a revised R1 (nR1), where j is the page of the lower-level write failure. Note that in contexts of  FIG. 12  such as associated with  FIG. 11 , if the finite field is a Galois Field the subtraction operation (“−”) illustrated in  FIG. 15C  is equivalent to a logical XOR operation. Other processing (not illustrated) is performed, in various embodiments, to store the page with the lower-level write failure (Pj), as well as the revised higher-level redundancy pages (nR0 and nR1). 
     Other than reception of write completion and status information in response to the commands sent as illustrated in  FIG. 15A , the processing illustrated in  FIG. 15B  is independent of processing illustrated in  FIG. 15A . In various scenarios, arrival order of the write completion and status information varies according to the NVM type, state, operating environment, and other factors, and in some circumstances is different than sending order or arrival order of the write commands corresponding to the arriving write completion and status information. As processing of  FIG. 15A  is independent of  FIG. 15B  (other than write completion and status information arrival being dependent on a corresponding write command being sent), in some embodiments and/or usage scenarios, write commands are being sent ( FIG. 15A ) while write completion and status information is being received/processed ( FIG. 15B ). 
     In some embodiments and/or usage scenarios, a significant latency transpires between write commands being sent to the NVMs and reception of write completion and status information from the NVMs in response to the write commands. In some embodiments and/or usage scenarios, write completion and status information other than in response to the commands sent as illustrated in  FIG. 15A  are provided by the NVMs intermixed with the write completion and status information provided in response to the commands sent as illustrated in  FIG. 15A , e.g. write completion and status information provided in response to write commands sent for other activities. 
     In various embodiments and/or usage scenarios, one or more pages that would otherwise be used for data information are unused. In various scenarios an unused page is a first, middle, or last page of a block, R-block, stripe, or sequence of pages any kind. In some circumstances, unused pages are unused a priori (“left out”), and in some circumstances unused pages are unused after some use (“removed from service”). An example of an unused page that is left out is a page that is manufactured incorrectly. An example of an unused page that is removed from service is a page that fails to write properly (e.g. as described as a lower-level write error with respect to  FIG. 15B ). Processing, e.g. relating to  FIGS. 10-12, 13A-13D, 14A-14B , and  15 A- 15 B, skips over any unused pages (whether left out or removed from service), such as by omitting the unused pages entirely from computations, or by performing computations as if all data on the unused pages were zero. 
     Fractional Higher-Level Redundancy 
     In the foregoing non-fractional RASIE techniques, storage capacity equivalent to an integral number of flash die (out of N total die) is dedicated to higher-level redundancy, enabling recovery from various types of errors. For example, out of 32 flash (e.g. NAND) die, one of the die is used for (non-fractional) RASIE-1 (e.g. RAID-5-like) redundancy, reducing capacity by 1/32 nd , but providing protection against various types of errors. 
     However, in some embodiments and/or usage scenarios, customers of storage systems prefer storage capabilities at various binary capacity points (for example 256 GB and 128 GB), and dedicating an entire flash die (or a plurality of entire flash die) to error recovery results in increased costs. Sometimes the increased costs are more than a linear increase based on additional flash die. For example, providing a 256 GB storage capability SSD is implementable using 32 8 GB NAND die total. If one of the NAND die is used for (non-fractional) RASIE-1, however, then only 31 of the NAND die remain for all other storage (user data plus any system data, such as mapping data, used by the SSD). In some implementations, using 33 NAND die increases cost non-linearly, e.g. an extra placement on a printed-circuit board, and further if the NAND die are in multi-die packages, such as two, four or eight die per package, then more than one die is used. 
     At a binary capacity of 256 GB, 256*10{circumflex over ( )}9 user-accessible bytes are provided, versus 256*2{circumflex over ( )}30 physical storage bytes provided by 328 GB NAND die, so about 7.37% of the physical storage bytes are available for other than user-accessible data, such as the system data. Dedicating one entire NAND die (of the 32 total) to higher-level redundancy consumes 1/32 of the physical storage bytes, or about 3.13%, leaving approximately 7.37%-3.13%=4.24% for the system data. If 4.24% is sufficient for the system data, then 32 8 GB NAND die are sufficient to implement the binary capacity of 256 GB while dedicating one of the die to higher-level redundancy. 
     At a binary capacity of 128 GB, implemented with 16 8 GB NAND die and dedicating one of the die to higher-level redundancy, approximately 7.37%-6.25%=1.12% of the physical storage bytes are available for system data. If 1.12% is insufficient for the system data, then additional NAND die are used (at additional cost) to implement the binary capacity of 128 GB while dedicating one of the die to higher-level redundancy. 
     In some embodiments and/or usage scenarios, a fractional higher-level redundancy technique reduces RASIE overhead to less than one entire flash die (e.g. a ‘fraction’ of a flash die) while covering various types of errors, such as errors as described with respect to the foregoing (non-fractional) RASIE-1, RASIE-2, and RASIE-3 techniques. The fractional higher-level redundancy techniques enable RASIE benefits for lower capacity points while retaining binary capacity. In some embodiments, fractional higher-level redundancy techniques include striping data across multiple blocks of flash devices (e.g. die). 
     For example, at a binary capacity of 128 GB, implemented with 16 8 GB NAND die, dedicating one-half of one of the die to higher-level redundancy consumes 1/32 of the physical storage bytes, or about 3.13%. Thus, approximately 7.37%-3.13%=4.24% remain available for the system data. If 4.24% is sufficient for the system data, then 16 8 GB NAND die are sufficient to implement the binary capacity of 128 GB while dedicating one-half of one of the die to higher-level redundancy. 
     Recall that a (non-fractional) RASIE-1 mode described elsewhere herein enables recovery from one error by using storage capacity equivalent to one flash die dedicated to higher-level redundancy information. In various embodiments, modes respectively termed (fractional) ‘RASIE-1/2’ and (fractional) ‘RASIE-1/4’ enable recovery from one error using respective storage capacities equivalent to one-half and one-fourth of one flash die dedicated to higher-level redundancy information. For example, RASIE-1/2 is used for 256 GB SSD and/or 128 GB SSD implementations. For another example, RASIE-1/4 is used for 64 GB SSD implementations. In various usage scenarios, RASIE-1/2 and/or RASIE-1/4 provide error protection for random failures (of, e.g., a read unit, a page, or a block) equivalent to RASIE-1. For example, in some scenarios, RASIE-1/2 for 256 GB and 128 GB SSDs, and RASIE-1/4 for 64 GB SSDs, provide error protection of random failures equivalent to protection provided by RASIE-1 for 512 GB SSDs (implemented with, e.g., 64 NAND flash die). Note that storage capacity dedicated to higher-level redundancy is not limited to power of two fractions (e.g. ½ and ¼ as in RASIE-1/2 and RASIE-1/4, respectively), as, for example, (fractional) ‘RASIE-1/3’ dedicates storage capacity approximately equal to ⅓ of a flash die to higher-level redundancy. 
       FIGS. 16A-16C  illustrate selected details of an embodiment of fractional higher-level redundancy.  FIG. 16A  serves as a key for  FIGS. 16B and 16C . As illustrated by the key information of  FIG. 16A , each square in  FIGS. 16B and 16C  represents a particular (dual-plane) block (e.g. Bk-1, Bk-2 B1, and B0 fork blocks per die) for a particular die (e.g. Dn-1, Dn-2 . . . D1, and D0 for n die total).  FIG. 16A  also illustrates a particular R-block having one block (the block) from each of the die. In  FIGS. 16B and 16C , blocks used for higher-level redundancy have a bold underlined font.  FIG. 16B  illustrates an example non-fractional RASIE embodiment, RASIE-1.  FIG. 16C  illustrates an example fractional RASIE embodiment, RASIE-1/2. 
     As described elsewhere herein, (non-fractional) RASIE-1 enables recovery from one error by using storage capacity equivalent to one flash die dedicated to higher-level redundancy information. In  FIG. 16B , the storage capacity equivalent to one flash die is illustrated by blocks having bold underline font text (Dn-1 Bk-1, Dn-1 Bk-2 Dn-1 B1, and Dn-1 B0). Thus the higher-level redundancy information is stored entirely in a single dedicated die, die Dn-1. In other embodiments (not illustrated), the higher-level redundancy information for RASIE-1 is stored in a different die (e.g. die Dn-2, blocks Dn-2 Bk-1 Dn-2 B0). In yet other embodiments (not illustrated), the higher-level redundancy information for RASIE-1 is stored in blocks of various die (e.g. blocks Dn-1 Bk-1, Dn-2 Bk-2 D0 B0). In yet other embodiments (not illustrated), the higher-level redundancy information is stored in blocks such that each R-block ‘contributes’ one block, with the one contributed block being any of the blocks in the R-block. For all of the aforementioned RASIE-1 embodiments, the higher-level redundancy information is shared across each R-block, as illustrated by the text bracket “RASIE Shared” bracketing each R-block in the figure. 
     As described elsewhere herein, (fractional) RASIE-1/2 enables recovery from one error by using storage capacity equivalent to one-half of one flash die dedicated to higher-level redundancy information. In  FIG. 16C , the storage capacity equivalent to one-half of one flash die is illustrated by blocks having bold underline font text (Dn-1 Bk-1 Dn-1 B3, and Dn-1 B1). Thus the higher-level redundancy information is stored entirely in one-half of a single dedicated die, every other block of die Dn-1. In other embodiments (not illustrated), the higher-level redundancy information for RASIE-1/2 is stored in every other block of die Dn-2 (e.g. blocks Dn-2 Bk-1 Dn-2 B3, and Dn-2 B1). In yet other embodiments (not illustrated), the higher-level redundancy information is stored in blocks such that each pair of R-blocks contributes one block, with the one contributed block being any of the blocks in the R-block pair. For all of the aforementioned RASIE-1/2 embodiments, the higher-level redundancy information is shared across each pair of R-blocks, as illustrated by the text bracket “RASIE Shared” bracketing each R-block pair in the figure. 
     In various embodiments and/or usage scenarios RASIE higher-level redundancy (e.g. RASIE-1 and RASIE-1/2) is used for higher-level error correction. When writing pages in each flash (e.g. NAND) die, a lower-level error-correcting code (such as a BCH or LDPC code) is used to protect data within individual pages of the flash die. The RASIE redundancy is an orthogonal, higher-level of redundancy applied across a number of pages (such as a specified number of pages from each of respective different flash die) to enable recovery from various error conditions. A first example of the various error conditions is a flash page that is uncorrectable using the lower-level redundancy because the flash page has accumulated too many errors for the lower-level redundancy to correct. A second example is a failed word line of one of the flash die that results in a portion (e.g. a page) of the flash die having the failed word line being inaccessible. A third example is a failed block of one of the flash die that results in a data of the failed block being inaccessible. A fourth example is an entire failed flash die that results in all data of the failed flash die being inaccessible. In various situations, the second, third, and fourth examples are physical failure mechanisms that are exemplary of hard (e.g. persistent) errors. 
     In some embodiments, RASIE (e.g. non-fractional RASIE such as RASIE-1 or fractional RASIE such as RASIE-1/2) uses an error-correcting code. In other embodiments, the RASIE redundancy uses an erasure-correcting code since a location of error(s) is known (e.g. the pages or portions thereof that failed to be corrected by lower-level redundancy). For example, a parity (XOR) code enables correction of one erasure using one redundant position out of N. RS codes enable correction of erasures, and an RS code using J redundant positions out of N enable correction of J erasures. Other erasure correcting techniques, such as those described in elsewhere herein are usable. 
     In some embodiments, RASIE redundancy information is stored by writing data in a “striped” fashion (e.g. as illustrated by Striping Direction  600  of  FIG. 6 ). One page from each flash (e.g. NAND) die is written in a determined order, wrapping around to write the next page in each die, until one block of each of the die has been written. E.g., the RASIE redundancy information is stored in a “die first, page second, block third” order that fills one block from each die before storing data into a second block in each die. 
     Because the aforementioned striping order writes an entire block in each die before writing to a second block in each die, block failures are managed, in some embodiments, with a full die of redundancy-one die out of N dedicated to storing the RASIE overhead. (Or equivalently, 1/Nth of the capacity distributed among the die.) 
     In various embodiments and/or usage scenarios, however, common forms of physical failures are not die failures-full die failures are perhaps 1/10th or less as likely as other physical failures (such as block or word line failures). 
     A problem with using one die out of N (or the equivalent in capacity distributed among the die) to store RASIE information for recovery of block failures is that overhead for the RASIE information is 1/N. In a small-capacity SSD, where N is small, the overhead becomes a higher percentage of the capacity, and the overhead plus system data is larger than the 2{circumflex over ( )}30/10{circumflex over ( )}9 factor that is the difference between flash storage capacity and binary capacity. 
     The inventor has realized that smaller capacity drives are less sensitive to die failures than larger capacity drives, as the smaller capacity drives have fewer flash die than the larger capacity drives. The inventor has further realized that a revised striping order is usable to protect against non-die physical failures without using excessive overhead. RASIE-1/2, as illustrated, e.g., by  FIG. 16C  is an example of the foregoing. 
     The revised striping order used for some implementations of fractional RASIE (e.g. RASIE-1/2 and RASIE-1/4) interleaves writing pages among a number of blocks in a group of blocks, so that the striping order in some embodiments is “die first, block within group of blocks, page, among groups of blocks.” For example, given a group of K blocks per each of N die, one page from each of the K blocks is written across all of the die (K*N pages total), and one (or more) of the pages contains RASIE higher-level redundancy information protecting the entire group of K*N pages (that forms one erasure-correcting codeword). Because no more than one page per block is used in the pages covered by the RASIE higher-level redundancy information (one of the erasure-correcting codewords), any single-block failure is still correctable. (Or, with more than a single-erasure-correcting code, more than a single-block failure is still correctable.) 
     In various embodiments, the pages within the K*N pages that form one erasure-correcting codeword are written in other orders, or multiple of the erasure-correcting codewords are written in an interleaved fashion. In some usage scenarios, a die-first order of writing enables higher performance. 
     In some embodiments, the value of K is an integer divisor of the number of blocks per plane of the flash (or blocks per die of the flash) so that all blocks within each of the die are used in a same manner. In further embodiments where the number of blocks per plane (or blocks per die) is a power of two or is divisible by a power of two, the value of K is a power of two. 
     In various embodiments, the value of K is adjusted based on capacity or other factors. For example, the product K*N is kept constant to maintain a same percentage overhead as a number of die is changed. For another example, the value of K*N is adjusted to achieve a desired degree of protection (an amount of data protected by an erasure-correcting code). 
     In various embodiments, RASIE redundancy being shared is a single erasure-correcting code (one portion out of N), or alternatively a J-erasure-correcting code (J portions out of N). Fractional RASIE increases the value of N to lower the overhead for any value of J. For example, by doubling the value of K, the value of J can be doubled without increasing the percentage overhead. This is sometimes preferable as, for example, the chance of getting more than two random errors (or erasures) out of 2*N pages is lower than the chance of getting two random errors (or erasures) out of just N pages. 
     In various embodiments, fractional RASIE (e.g. RASIE-1/2 and RASIE-1/4) higher-level redundancy information computation is performed similarly to non-fractional RASIE (e.g. RASIE-1) higher-level redundancy computation, except that multiple data blocks within a particular die are are covered by each block of higher-level redundancy information. See, for example,  FIGS. 10-12 , and associated description elsewhere herein. In various embodiments, fractional RASIE higher-level redundancy error recovery is performed similarly to non-fractional RASIE higher-level redundancy error recovery, except that more data blocks are covered by each block of higher-level redundancy computation. See, for example,  FIGS. 13A-D ,  14 A-B, and  15 A-C, and associated description elsewhere herein. 
     Example Implementation Techniques 
     In some embodiments, various combinations of all or any portions of operations performed by a system implementing fractional higher-level redundancy for NVMs (e.g. flash memories, such as NAND flash memories), a computing-host flash memory controller, and/or an SSD controller (such as SSD Controller  100  of  FIG. 1A ), and portions of a processor, microprocessor, system-on-a-chip, application-specific-integrated-circuit, hardware accelerator, or other circuitry providing all or portions of the aforementioned operations, are specified by a specification compatible with processing by a computer system. The specification is in accordance with various descriptions, such as hardware description languages, circuit descriptions, netlist descriptions, mask descriptions, or layout descriptions. Example descriptions include: Verilog, VHDL, SPICE, SPICE variants such as PSpice, IBIS, LEF, DEF, GDS-II, OASIS, or other descriptions. In various embodiments, the processing includes any combination of interpretation, compilation, simulation, and synthesis to produce, to verify, or to specify logic and/or circuitry suitable for inclusion on one or more integrated circuits. Each integrated circuit, according to various embodiments, is designable and/or manufacturable according to a variety of techniques. The techniques include a programmable technique (such as a field or mask programmable gate array integrated circuit), a semi-custom technique (such as a wholly or partially cell-based integrated circuit), and a full-custom technique (such as an integrated circuit that is substantially specialized), any combination thereof, or any other technique compatible with design and/or manufacturing of integrated circuits. 
     In some embodiments, various combinations of all or portions of operations as described by a computer readable medium having a set of instructions stored therein, are performed by execution and/or interpretation of one or more program instructions, by interpretation and/or compiling of one or more source and/or script language statements, or by execution of binary instructions produced by compiling, translating, and/or interpreting information expressed in programming and/or scripting language statements. The statements are compatible with any standard programming or scripting language (such as C, C++, Fortran, Pascal, Ada, Java, VBscript, and Shell). One or more of the program instructions, the language statements, or the binary instructions, are optionally stored on one or more computer readable storage medium elements. In various embodiments, some, all, or various portions of the program instructions are realized as one or more functions, routines, sub-routines, in-line routines, procedures, macros, or portions thereof. 
     CONCLUSION 
     Certain choices have been made in the description merely for convenience in preparing the text and drawings, and unless there is an indication to the contrary, the choices should not be construed per se as conveying additional information regarding structure or operation of the embodiments described. Examples of the choices include: the particular organization or assignment of the designations used for the figure numbering and the particular organization or assignment of the element identifiers (the callouts or numerical designators, e.g.) used to identify and reference the features and elements of the embodiments. 
     The words “includes” or “including” are specifically intended to be construed as abstractions describing logical sets of open-ended scope and are not meant to convey physical containment unless explicitly followed by the word “within.” 
     Although the foregoing embodiments have been described in some detail for purposes of clarity of description and understanding, the invention is not limited to the details provided. There are many embodiments of the invention. The disclosed embodiments are exemplary and not restrictive. 
     It will be understood that many variations in construction, arrangement, and use are possible consistent with the description, and are within the scope of the claims of the issued patent. For example, interconnect and function-unit bit-widths, clock speeds, and the type of technology used are variable according to various embodiments in each component block. The names given to interconnect and logic are merely exemplary, and should not be construed as limiting the concepts described. The order and arrangement of flowchart and flow diagram process, action, and function elements are variable according to various embodiments. Also, unless specifically stated to the contrary, value ranges specified, maximum and minimum values used, or other particular specifications (such as flash memory technology types; and the number of entries or stages in registers and buffers), are merely those of the described embodiments, are expected to track improvements and changes in implementation technology, and should not be construed as limitations. 
     Functionally equivalent techniques known in the art are employable instead of those described to implement various components, sub-systems, operations, functions, routines, sub-routines, in-line routines, procedures, macros, or portions thereof. It is also understood that many functional aspects of embodiments are realizable selectively in either hardware (e.g., generally dedicated circuitry) or software (e.g., via some manner of programmed controller or processor), as a function of embodiment dependent design constraints and technology trends of faster processing (facilitating migration of functions previously in hardware into software) and higher integration density (facilitating migration of functions previously in software into hardware). Specific variations in various embodiments include, but are not limited to: differences in partitioning; different form factors and configurations; use of different operating systems and other system software; use of different interface standards, network protocols, or communication links; and other variations to be expected when implementing the concepts described herein in accordance with the unique engineering and business constraints of a particular application. 
     The embodiments have been described with detail and environmental context well beyond that required for a minimal implementation of many aspects of the embodiments described. Those of ordinary skill in the art will recognize that some embodiments omit disclosed components or features without altering the basic cooperation among the remaining elements. It is thus understood that much of the details disclosed are not required to implement various aspects of the embodiments described. To the extent that the remaining elements are distinguishable from the prior art, components and features that are omitted are not limiting on the concepts described herein. 
     All such variations in design are insubstantial changes over the teachings conveyed by the described embodiments. It is also understood that the embodiments described herein have broad applicability to other computing and networking applications, and are not limited to the particular application or industry of the described embodiments. The invention is thus to be construed as including all possible modifications and variations encompassed within the scope of the claims of the issued patent.