Patent Publication Number: US-7721140-B2

Title: Systems and methods for improving serviceability of a memory system

Description:
BACKGROUND OF THE INVENTION 
     This invention relates generally to computer systems, and more particularly to improving the serviceability of a memory system in the presence of defective memory parts. 
     Contemporary high performance computing main memory systems are generally composed of one or more dynamic random access memory (DRAM) devices, which are connected to one or more processors via one or more memory control elements. Overall computer system performance is affected by each of the key elements of the computer structure, including the performance/structure of the processor(s), any memory cache(s), the input/output (I/O) subsystem(s), the efficiency of the memory control function(s), the main memory device(s), the type and structure of the memory interconnect interface(s) and the type and efficiency of any failure detection/correction function associated with one or more elements of the system. 
       FIG. 1  relates to U.S. Pat. No. 5,513,135 to Dell et al., of common assignment herewith, and depicts an early synchronous memory module. The memory module depicted in  FIG. 1  is a dual in-line memory module (DIMM). This module is composed of synchronous DRAMs  108 , buffer devices  112 , an optimized pinout, and an interconnect and capacitive decoupling method to facilitate high performance operation. The patent also describes the use of clock re-drive on the module, using such devices as phase-locked loops (PLLs). 
       FIG. 2  relates to U.S. Pat. No. 6,173,382 to Dell et al., of common assignment herewith, and depicts a computer system  210  which includes a synchronous memory module  220  that is directly (i.e. point-to-point) connected to a memory controller  214  via a bus  240 , and which further includes logic circuitry  224  (such as an application specific integrated circuit, or “ASIC”) that buffers, registers or otherwise acts on the address, data and control information that is received from the memory controller  214 . The memory module  220  can be programmed to operate in a plurality of selectable or programmable modes by way of an independent bus, such as an inter-integrated circuit (I2C) control bus  234 , either as part of the memory initialization process or during normal operation. When utilized in applications requiring more than a single memory module connected directly to a memory controller, the patent notes that the resulting stubs can be minimized through the use of field-effect transistor (FET) switches to electrically disconnect modules from the bus. 
     Relative to U.S. Pat. No. 5,513,135, U.S. Pat. No. 6,173,382 further demonstrates the capability of integrating all of the defined functions (address, command, data, presence detect, etc) into a single device. The integration of functions is a common industry practice that is enabled by technology improvements and, in this case, enables additional module density and/or functionality. 
     Extensive research and development efforts are invested by the industry, on an ongoing basis, to create improved and/or innovative solutions to maximizing overall system performance and density by improving the memory system/subsystem design and/or structure. The need for high-availability and minimal (if any) downtime presents further challenges as related to overall system reliability due to customer expectations that new computer systems will markedly surpass existing systems in regard to mean-time-between-failure (MTBF), in addition to offering additional functions, increased performance, increased storage, lower operating costs, etc. 
     The use of enhanced error correction techniques has been a major factor in improving MTBF, however it is becoming increasingly difficult to identify a single failing module within a cluster (2, 4 or more) of modules that are accessed in parallel to service a cache line access (typically 64 bytes, 128 bytes or larger). This is particularly true in cases where the error is identified by the ECC structure as being “uncorrectable.” During scheduled or unscheduled repair actions, given the limited amount of failure data (especially in cases where an uncorrectable error has resulted in the repair action) and the need to bring the system online quickly, it is common for more than one memory module to be removed in response to an apparent memory system failure. It would be desirable to have the capability of quickly and accurately identifying specific failing memory modules in order to reduce the number of functional modules that are unnecessarily replaced. 
     BRIEF SUMMARY OF THE INVENTION 
     An exemplary embodiment includes a method for identifying a failing memory element in a memory system when two or more modules operate in unison in response to a read request. The method includes receiving syndrome bits and an address associated with an uncorrectable error (UE). In response to a previous correctable error (CE) having occurred, the location of the previous CE is retrieved. The location of the CE specifies a memory device position of the CE. A location of the UE is determined using the location of the previous CE and the syndrome bits of the UE as input. The location of the UE specified a memory device position. A failing memory element associated with the location of the UE is identified. 
     An exemplary embodiment also includes a method for identifying a failing memory element in a memory system where two or more memory modules operate in unison in response to a read request. The method includes receiving an address associated with an UE and data related to the UE including one or more of syndrome bits and read data related to the UE. In response to a previous CE having occurred, data related to the previous CE is retrieved. The failing memory element is identified using the data related to the previous CE and the data related to the UE as input. 
     A further exemplary embodiment includes a memory system for identifying a failing memory element. The memory system includes two or more memory modules, storage means and a processor. Each of the memory modules includes memory devices and at least two of the two or more memory modules operate in unison in response to read requests. The storage means stores failure information associated with a first fail at an address. The processor includes logic for facilitating receiving syndrome bits and an address associated with an UE. In response to a previous CE having occurred, the location of the previous CE is retrieved. The location of the CE specifies a memory device position of the CE. A location of the UE is determined using the location of the previous CE and the syndrome bits of the UE as input. The location of the UE specified a memory device position. A failing memory element associated with the location of the UE is identified. 
     Other systems, methods, and/or computer program products according to embodiments will be or become apparent to one with skill in the art upon review of the following drawings and detailed description. It is intended that all such additional systems, methods, and/or computer program products be included within this description, be within the scope of the present invention, and be protected by the accompanying claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Referring now to the drawings wherein like elements are numbered alike in the several FIGURES: 
         FIG. 1  depicts an exemplary early synchronous memory module; 
         FIG. 2  depicts an exemplary computer system with a fully buffered synchronous memory module that is directly connected to a memory controller; 
         FIG. 3  depicts an exemplary memory system, shown with a single, traditional multi-drop stub bus; 
         FIG. 4  depicts a fully buffered synchronous memory module and system structure, where the fully buffered synchronous memory module includes a repeater function; 
         FIG. 5  depicts a block diagram of a computer memory system which includes multiple independent cascade interconnect memory interface busses that operate in unison to support a single data access request; 
         FIG. 6  depicts a block diagram of a computer memory system that may be utilized to implement exemplary embodiments of the present invention; 
         FIG. 7  depicts a memory structure with cascaded memory modules and unidirectional busses that may be utilized to implement exemplary embodiments of the present invention; 
         FIG. 8  is a process flow that may be implemented by exemplary embodiments of the present invention to identify the location of uncorrectable errors; 
         FIG. 9  depicts a memory structure with cascaded memory modules and unidirectional busses that may be utilized to implement exemplary embodiments of the present invention; and 
         FIGS. 10-13  depict exemplary circuits that may be implemented by exemplary embodiments of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     Exemplary embodiments assist in the identification of specific failing memory modules in a memory system in order to reduce the number of memory modules being replaced when a failure occurs in the memory system. The location information of a first, correctable, byte error is utilized to find the location of a second, uncorrectable, byte error. In exemplary embodiments, the location of the second error is employed to identify a defective memory element (e.g., module or memory device). In addition, the second byte error may be corrected, thereby allowing the memory system to continue execution for production and/or diagnostic purposes. 
     Current state-of-the-art in some computer systems is to encode the data using a one byte error correct, two byte error detect Reed-Solomon (or equivalent) error control code. This code allows the system to recover from single bit upsets as well as failures affecting up to an entire DRAM memory chip when the code is applied appropriately (e.g., so that any memory chip failure affects at most one byte). This first error that is corrected is referred to as a correctable error (CE). The class of memory chip failures includes massive failures (e.g., a “chipkill”) or more commonly, failures of individual I/O lanes caused by events such as I/O driver/receiver failures, and connector failures. 
     When a second error takes place when using the above Reed-Solomon code, the code (as it is currently being employed) cannot correct the error and generates an uncorrectable error (UE) signal. A common policy is to halt execution of the system when an UE occurs. Nevertheless, it is frequently the case that the two errors (i.e., the CE and the UE) do not initially occur simultaneously—rather, a first failure type occurs at an address (e.g. a single bit failure, an I/O failure, a chip kill, etc) and only after some additional time, a second failure occurs which affects a different bit, byte or portion of a byte within the same ECC word. In the intervening time, additional accesses to the data associated with the address (e.g. accesses requested by the processor, accesses due to a scrubbing mechanism, etc) will result in persistent errors coinciding with the first failure type and location. 
       FIGS. 3-5  that follow are used as a means of describing example conventional memory structures to which exemplary embodiments of the present invention would apply. 
       FIG. 3 , from U.S. Pat. No. 6,510,100 to Grundon et al., of common assignment herewith, depicts a simplified diagram and description of a memory system  310  that includes up to four registered DIMMs  340  on a traditional multi-drop stub bus. The subsystem includes a memory controller  320 , an external clock buffer  330 , registered DIMMs  340 , an address bus  350 , a control bus  360  and a data bus  370  with terminators  395  on the address bus  350  and the data bus  370 . Although only a single memory channel is shown in  FIG. 3 , systems produced with these modules often included more than one discrete memory channel from the memory controller, with each of the memory channels operated singly (when a single channel was populated with modules) or in parallel (when two or more channels where populated with modules) to achieve the desired system functionality and/or performance. Using conventional registered DIMMs, a data bus width of 72 bits would be available from each memory channel. Although only one channel is shown, a system would often have 2, 4 or more memory channels of the type shown, resulting in a transfer of 32, 64, 128 or more bytes of data in response to a cache line operation. Further, each DIMM  340 , or memory module, would generally include 72 bits of data, permitting the use of ECC to correct most data errors, with the ECC circuitry, or logic, located in the memory controller  320 . 
       FIG. 4 , from U.S. Pat. No. 6,587,912 to Bonella et al., depicts a synchronous memory module  410  and system structure in which the repeater hubs  420  include local re-drive of the address, command and data to the local memory devices  401  and  402  via buses  421  and  422 ; generation of a local clock (as described in other figures and the patent text); and the re-driving of the appropriate memory interface signals to the next module or component in the system via bus  400 . 
       FIG. 5  depicts a memory system composed of an integrated processor chip  500 , which contains one or more processor elements and an integrated memory controller  510 . In the configuration depicted in  FIG. 5 , multiple independent cascade interconnected memory interface busses  506  are logically aggregated together to operate in unison to support a single independent access request at a higher bandwidth with data and error detection/correction information distributed or “striped” across the parallel busses and associated devices. The memory controller  510  attaches to four narrow/high speed point-to-point memory busses  506 , with each bus  506  connecting one of the several unique memory controller interface channels to a cascade interconnect memory subsystem  503  which includes at least a hub device  504  and one or more memory devices  509  (or DRAMs). Using contemporary synchronous memory devices, an access initiated by the memory controller  510  would commonly result in a transfer of 128 bytes of data (8 byte DIMM width×4 DIMMs operating in unison×4 bit burst from each selected DRAM). 
     Using conventional buffered DIMMs, a data bus width of 72 bits would be available on each DIMM (typically by accessing 9 memory devices  509  that are 8 bits wide, or by accessing 18 memory devices  509  that are 4 bits wide), permitting the use of ECC to identify and correct most data errors. As with  FIGS. 3 and 4 , the data path ECC circuitry, or logic in  FIG. 5 , would generally be located in the memory controller  510  and include the check bit generation and subsequent error detection and correction. 
     Although contemporary memory ECC structures may be implemented as described relative to  FIGS. 3-5 , the location of the ECC function (e.g., the ECC circuitry or logic) may reside outside of the memory controller in other system elements, including the processor(s) themselves, the memory hub, register, buffer or other memory interface device residing between the memory controller and the memory devices, or in the memory devices themselves. 
       FIG. 6  depicts a block diagram of a computer memory system that may be utilized to implement exemplary embodiments of the present invention.  FIG. 6  depicts a memory system composed of an integrated processor chip  600 , which contains one or more processor elements and an integrated memory controller  610 . In the configuration depicted in  FIG. 6 , multiple independent cascade interconnected memory interface busses  606  are logically aggregated together to operate in unison to support a single independent access request at a higher bandwidth with data and error detection/correction information distributed or “striped” across the parallel busses and associated devices. The memory controller  610  attaches to four narrow/high speed point-to-point memory busses  606 , with each bus  606  connecting one of the several unique memory controller interface channels to a cascade interconnect memory subsystem  603  which includes at least a hub device  604  and one or more memory devices  609  (or DRAMs). 
     The memory system in  FIG. 6  is similar to the memory system depicted in  FIG. 5  with the addition of a service processor  612 , non-volatile storage (NVS)  614  for storing information about the errors (e.g., their address, their location, the data retrieved, and syndrome bits), and an ECC logic block  616 . In the exemplary embodiment depicted in  FIG. 6 , portions of the UE processing described herein (e.g., see  FIG. 8 ) are located within the ECC logic  616  in the memory controller and other portions of the UE processing are located in the service processor  612 . An advantage to utilizing the service processor  612  external to the integrated processor chip  600  is that it may be utilized to perform error diagnostics even when the integrated processor chip  600  is not operational. Similarly, an advantage to utilizing the service processor  612  external to the memory controller  610  is that it may be utilized to perform error diagnostics even when the memory controller  610  is not operational. In addition, an advantage to using NVS  614  for storing information about the errors is that it is accessible when the memory controller  610  and/or the service processor is not operational. 
       FIG. 7  depicts a memory structure with cascaded memory modules and unidirectional busses that may be utilized to implement exemplary embodiments of the present invention.  FIG. 7  is another view of  FIG. 6 , with the service processor  612  and NVS  614  located external to the memory controller  610 . In the example described herein, and typical for current memory systems, the cache line is 128 bytes; however this number could be varied without departing from the scope of exemplary embodiments of the present invention described herein. The memory system depicted in  FIG. 7  includes only one memory module  702 , with the option of having one or more additional modules cascade interconnected to one or more of the modules  702  (as shown, for example in  FIG. 6 ). Other embodiments may include only one rank of memory modules or other combinations of modules as required to optimally provide the required system line size (e.g. 32, 64, 128, 256 bytes, etc). 
     As depicted in  FIG. 7 , the memory controller  610  attaches to four narrow/high speed point-to-point memory busses  606   a ,  606   b ,  606   c ,  606   d  with each bus connecting one of the several unique memory controller interface channels to a memory module  702   a ,  702   b ,  702   c  and  702   d , respectively. In the exemplary embodiment depicted in  FIG. 7 , each of the memory modules  702  contains memory devices  509 , each with 8 data input/output (I/O) lanes and operating with a burst length of four. The burst length is the number of data beats (e.g. sequential data transfers) that each of the selected memory devices  509  will process (e.g. read or write) through the I/O lanes (via the memory busses  606 ) in any given access to the memory device(s)  509 . In the exemplary embodiment, a total of 32 memory devices  509  are needed to service a 128 byte cache line. Generally, as depicted in  FIG. 7 , an additional, ninth, memory device  509  is provided for every eight memory devices  509  in the memory module  702 . In the example depicted in  FIG. 7  this provides a total of four additional memory devices  509  across the four memory modules  702 . In the exemplary embodiment, one of the four additional memory devices  509  is designated as a spare memory device  509 , with the other three being utilized to store ECC information. 
     A Reed-Solomon code that protects 32 bytes of data with 3 additional redundant bytes is applied on each memory device data beat. This code is capable of correcting any one byte error and detecting any two byte errors. When two byte errors occur, this Reed Solomon code may be utilized to correct both errors if the location of both of the errors is known. In the memory system depicted in  FIG. 7 , an up to one byte error will be identified and corrected on any of the memory devices  509  on one of the four memory modules  702 . Exemplary embodiments of the present invention are directed to determining the location of a second error, affecting up to a full byte of data (e.g. a chip kill). Once the location of the second error is determined, the second error may be corrected and/or the location may be determined and reported to aid in identifying which memory element(s) (e.g., memory device  509 , memory subsystem  603 , memory module  702 , etc) have failed, and which of the replaceable elements (e.g. memory module(s), memory card(s), etc) should be replaced to correct the error(s) in the memory system. 
     In exemplary embodiments of the present invention, the memory devices  509  are numbered sequentially from 1 to 36 starting at the leftmost memory device  509  in  FIG. 7  to designate a location of the memory device  509  (and therefore to designate an error location). The nine memory devices  509  in memory module  702   a  are numbered from 1 to 9; the memory devices  509  in memory module  702   b  from 10 to 18; the memory devices  509  in memory module  702   c  from 19 to 27; and the memory devices  509  in memory module  702   d  from 28 to 36. Thus, if an error is in location  20 , the error is associated with the second memory device  509  from the left in memory module  702   c . Using the device position, or location, a corresponding failing memory element, such as memory modules  702   c , can be identified for replacement. The failure location information can further be used to facilitate subsequent repair of the failing module, by including the device failing location(s), along with the failing module, when returning the module for repair. In further exemplary embodiments, more or less than 9 memory devices may be installed on memory modules  702  (typically 18, 36 or 72), with the failing bit, bits or byte location used to identify the failing module (e.g. module  702   c  having 18, 8 bit wide memory devices and the error(s) related to the 4 th  group of 8 bits across the 72 bit module data width). In multi-rank (memory modules, such as the 72 bit module described having 18, 8 bit wide memory devices, the address of the failure would identify the failing memory device rank. 
     The memory module configuration (e.g., memory device count, memory device data width, module density, etc) is determined during system initialization, at which point the processor, service processor, memory controller or other control element interrogates a non-volatile memory (e.g. the Serial Presence Detect, or SPD EEPROM) installed on the memory modules, generally by means of an IIC or similar interface bus. Once the memory subsystem structure is determined, the memory interface is initialized to permit high speed communication (e.g., via busses  606 ) and further initialization and verification operations are completed prior to normal system operation. During these subsequent steps, the ECC logic  616  and non-volatile storage  614  can be configured to permit the correct identification of failing memory devices and/or modules, based on the memory device and module information determined during the module interrogation process. 
     In exemplary embodiments of the present invention, when a UE or CE is detected by the standard ECC processing (e.g., the standard ECC decoder for detecting the existence of double errors and correcting single errors that is already being performed by contemporary memory systems), information about the errors is stored in a storage medium (e.g., the NVS  614 ) accessible by another system, such as the service processor  612 . In the exemplary embodiment, this second system analyzes the information through a software and/or hardware tool that implements an algorithm to locate the second, UE, error. Other exemplary embodiments may complete this analysis in the processor, the memory controller, an external processing device or some other processing or analysis means. In order to determine the location of the failures contributing to the UE error, the uncorrectable error has to occur at an address for which a correctable error was previously logged during a memory access. The minimum necessary information needed by the second system is the syndromes of the data read for the UE and the location of the (persistent) first error, or CE. The syndromes of the UE are obtained by multiplying the ECC parity check matrix (usually denoted by H) by the received data vector (usually denoted by c, with c being a column vector). This can be expressed as s=Hc. The syndromes of the UE together with the location of the first error are all that is needed by the software tool in the second system to obtain the location of the second error, and if desired, the magnitudes of both the first and second errors. Alternatively, the actual data read when the UE and CE occur may be stored instead of or in addition to the syndromes. This may allow the software tool on the second system to determine the nature of the failure (e.g., that an I/O lane always outputs a “1” bit regardless of the input value). This information can also be used, for example, to gather failure statistics. 
       FIG. 8  is a process flow that may be implemented by exemplary embodiments of the present invention to identify the location of an UE (e.g., a second byte error).  FIG. 8  depicts an exemplary process for identifying a failing memory element (e.g., a memory device and/or memory module) in a memory system (see, for e.g.,  FIGS. 5-7  and  9 ) when two or more memory modules operate in unison in response to a read request. At block  802 , an error is detected when an access is made to a particular storage location. In exemplary embodiments, the error is detected by the standard ECC processing. If the error is a CE, as determined at block  804 , then block  806  is performed and the standard ECC CE processing is performed to determine the location of the CE and to correct the CE. In exemplary embodiments block  806  is performed by the standard ECC processing located in contemporary memory controllers. Next, block  808  is performed to store the address and location of the CE. In exemplary embodiments, this information is stored in NVS  614  so that it may be accessed even when the memory controller  610  is not operational. In further exemplary embodiments, only a single copy of each unique failure will be stored, and an error count will be compiled for each unique fail (up to a pre-determined maximum count), which can be used to verify that the CE is a persistent error. 
     If it is determined at block  804  that the error is a UE, then block  810  is performed. At block  810 , the address and syndrome bits of the UE are stored, for example, in the NVS  614 . At block  812 , a check is made to determine if a CE has previously occurred. IF a CE has not previously occurred, then block  814  is executed to perform standard UE processing. In exemplary embodiments this is performed by the standard ECC processing in contemporary memory controllers and may result in the memory system operation being halted. The processing depicted in  FIG. 8  may be utilized in a chip-kill situation, in that when a double error occurs there is a good chance that the one of the errors came from the chip-kill. 
     If a CE has previously occurred, then block  816  is performed to determine the failing memory element associated with the UE. In exemplary embodiments, this is performed using the syndrome bits of the UE and the location of the previous CE. In exemplary embodiments, this is performed by assuming that the location (e.g., the bit location or the byte location) for one of the errors contributing to the UE is the same as the location of the error(s) for the CE. In exemplary embodiments the location specifies a position of a memory device on a memory module. In alternate exemplary embodiments, more elaborate schemes are utilized to determine the location of the UE, such as the one described below herein. Based on the location of the UE, the failing memory element is identified. Block  820  is performed if the memory controller  610  is not in diagnostic mode, as determined at block  818 . At block  820 , the location of the UE is reported (e.g. the one or more modules contributing one or more bit failures at the time the UE was identified). In alternate exemplary embodiments, the failing module location will consist of the module having the first, persistent CE error—with the module having the second error (if on a separate module) not identified until a CE error threshold is met, or until the second error is identified as a first, persistent error during a future access which includes an uncorrectable error. The reporting can include initiating a visual indicator (e.g., a flashing light or LED, a pattern of flashing lights) on the memory module, on the memory socket (into which the memory module is installed), next to the failing module, etc, and/or reporting the error via an alert, printed output, electronic notification or other verbal, visual or related means. Otherwise, if the memory controller  610  is in diagnostic mode, block  822  is performed. At block  822 , the UE is corrected and the memory subsystem continues to operate. Block  822  can be applied when the system is running in a diagnostics mode, where after correcting the second error, storing the information in NVS  614  and halting, the system is turned back on again (in spite of the double failure) and operated for diagnostics and repair purposes. In addition, the diagnostics mode may have applications in production environments where actual system up-time is more important than exposure to triple errors (e.g., massive search systems where incorrect calculations are insignificant if they happen with a very low probability). As discussed previously, once the location of the failures contributing to the uncorrectable error condition are known, general techniques for implementing the required ECC decoder to correct for the second error (i.e., the UE) are known in the art. 
     In alternate exemplary embodiments, the data read from the memory devices when the CE occurred is stored along with the address of the CE at block  808 . In addition, at block  810 , the address of the UE and data that includes one or both of the syndrome bits and data read from the memory device when the UE occurred is stored in the storage device. At block  816 , the read data related to the previous CE and the read data related to the UE are used to identify the failing memory element. 
     In further alternate exemplary embodiments, block  812  determines if a previous CE has occurred at the same address as the current UE. Only if a CE has occurred at the same address as the UE, are blocks  816 - 822  performed. If a CE has not occurred at the same address as the UE, then block  814  is performed. 
     Portions or all of the processing depicted in  FIG. 8  may be implemented by ECC logic  616  that contains standard ECC processing for the memory controller  610  with extensions to implement the processing described herein to determine the location of a second byte error, or UE. In addition, the processing depicted in  FIG. 8  may be implemented in hardware, software, or a combination of hardware and software. 
     In exemplary embodiments, portions or all of the processing depicted in  FIG. 8  is implemented by a service processor  612  to allow the processing to take place when the memory controller  610  is not functional and/or to allow the processing to take place without impacting the performance of the standard ECC processing in the ECC logic  616 . For example, the standard ECC processing in the ECC logic  616  may already detect errors and write information about errors (both CE and UE) to the NVS  614  (e.g., perform blocks  802 ,  808  and  810 ). When it detects an UE, the standard ECC processing may be modified to invoke the service processor  612  to perform the processing in  FIG. 8  starting at block  812  to determine if a previous CE has occurred. If it has not, then control may return to the standard ECC processing to perform block  814 . If a previous CE has occurred, then the service processor  612  executes blocks  816  through  822  (possibly by calling modules in the standard ECC processing if they apply). The previous example is just one of ways that the processing depicted in  FIG. 8  may be split up across the ECC logic  616  and the service processor  612 . 
     In further exemplary embodiments, upon completion of block  816  (or  810  or  822 ), the service processor, processor or memory controller may invoke one or more methods such as chip sparing, bit lane sparing, or other available methods to correct the persistent error(s) and/or the new error which contributed to the uncorrectable error condition, in an effort to prevent future UE events. Once the corrective actions are taken and verified, the stored information associated with the repaired failure(s) would be cleared and potentially stored in another region of the NVM for failure tracking and/or other reporting purposes unrelated to subsequent uncorrectable error handling. 
       FIG. 9  depicts a memory structure with cascaded memory modules and unidirectional busses that may be utilized to implement exemplary embodiments of the present invention. The memory system depicted in  FIG. 9  is similar to the system depicted in  FIG. 7  except that all of the processing depicted in  FIG. 8  is contained in the ECC logic block  906  and at least a portion of the ECC logic block  906  related to UE recovery (e.g. blocks  810  and above) is located in the processor  904 . The information utilized by the processing depicted in  FIG. 8  is stored in NVS  614 . 
     The exemplary embodiment depicted in  FIG. 9  includes a memory system with two or more memory modules  702  operating in unison in response to read requests from the memory controller  902 . The memory modules  702 , as depicted in  FIG. 9  each include one or more memory devices. The memory system also includes a storage means (e.g., non-volatile storage  614  or other storage located in a processor  904  or ECC logic  906 ) for storing failure information associated with a first fail at an address. The memory system also includes a processor  904  (e.g., all or a subset of the processor  904  and/or the service processor  612  depicted in  FIG. 6 ) for facilitating logic (contained for e.g., in the ECC logic block  906 ). The logic facilitates identifying a failing memory element (e.g., a memory device on a memory module  702  or a memory module  702 ) associated with the location of a UE. 
     An exemplary embodiment of the present invention described below herein includes a decoding procedure for correcting two errors where the location of the first error is known. Exemplary embodiments may be utilized to determine defective memory devices, DIMMs (or memory modules), etc., in a system where 4 DIMMs (or some other number of DIMMs), operating in unison, service any one line request. The code described below is capable of correcting any one byte error and detecting any two byte errors. When a persistent, CE is found, information about the CE is stored (e.g., address, location, data accessed, syndrome bits) and is subsequently used, in response to a UE, to locate and optionally to correct a second error which occurred in the same ECC word (e.g., memory access) with the persistent CE to result in an uncorrectable error condition. The information related to the second error is then used, in conjunction with the information related to the first error, to locate the one or more defective DIMM(s), or memory module  702 . 
     The exemplary embodiment described below is implemented in hardware, however, for the purposes of the task above, it is in principle not necessary to implement this feature in the actual hardware; it will suffice to store the code&#39;s syndrome information along with the first error location in a non-volatile memory so that the information can be retrieved at repair time and then analyzed with a software tool. On the other hand, the exemplary embodiments that can determine the location of a UE error may be used to increase the actual error correction capacity of the system (with the caveat of the associated exposure to triple errors), in which case it may be preferred, e.g., for performance reasons, to implement it in hardware. Other implementation options exist: for example, it is feasible to provide the hardware with the functionality to find the location of the UE but not correct it. 
     In exemplary embodiments, the algorithm described below is used only when the standard ECC decoder for detecting double errors and correcting single errors signals that it has detected an UE. Those UEs consisting of exactly two errors will be corrected if the location of one of the errors is loaded in the appropriate input variable. This algorithm too will have an UE signal (the true one) in case the error cannot be corrected by it. 
     A Reed Solomon code for detecting two errors and correcting a single error generally utilizes an ECC parity check matrix or H-matrix to represent the parity-check equations that each ECC word must satisfy. The H-matrix is also used to specify the XOR logic for the generation of ECC check bits and syndrome bits. The result of the XOR of those bits of an ECC word corresponding to the ones in a row vector of the H-matrix is always equal to 0. Mathematically, if  V  is a binary vector representing an ECC word, then the product of the ECC matrix H and the transpose of  V  is an all zeros vector, i.e., H· V   t =0 mod 2. From the parity-check equations, the values of check bits can be calculated from the data bits through XOR operations. The data bits together with the corresponding check bits form an ECC word to be stored in the memory. On the other hand, when an ECC encoded word is fetched from the memory, the parity-check equations are used to generate the syndrome that represents the validity of the received data. 
     For example, let p(x)=1+x+x 6 +x 7 +x 8 , a binary primitive polynomial, and α be a root of p(x) in the finite field GF(2 8 ) of 256 elements. The companion matrix of α is: 
     
       
         
           
             T 
             = 
             
               [ 
               
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
                 
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
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                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
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                     1 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     Certain powers of T will be used to represent the elements of the ECC H-matrix. The powers T i  are companion matrices of the non zero finite field elements α i , 0≦i≦254. In addition, each α i  can be expressed as an 8-bit binary vector. 
     A (288, 264) SSC-DSD (single symbol error correcting—double symbol error detecting) code with 24 check bits (3 symbols) and 264 data bits (33 symbols) for a total of 288 bits (36 symbols) is constructed and specified by the H-matrix of the form: 
     
       
         
           
             
               H 
               = 
               
                 [ 
                 
                   
                     M 
                     1 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     M 
                     2 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     I 
                     24 
                   
                 
                 ] 
               
             
             , 
             
               
 
             
             ⁢ 
             
               where 
               ⁢ 
               
                   
               
               ⁢ 
               
                 M 
                 1 
               
               ⁢ 
               
                   
               
               ⁢ 
               is 
             
           
         
       
       
         
           
              
             
               
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   
                     T 
                     2 
                   
                 
                 
                   
                     T 
                     49 
                   
                 
                 
                   
                     T 
                     50 
                   
                 
                 
                   
                     T 
                     51 
                   
                 
                 
                   
                     T 
                     227 
                   
                 
               
               
                 
                   I 
                 
                 
                   T 
                 
                 
                   
                     T 
                     2 
                   
                 
                 
                   
                     T 
                     3 
                   
                 
                 
                   
                     T 
                     4 
                   
                 
                 
                   
                     T 
                     156 
                   
                 
                 
                   
                     T 
                     157 
                   
                 
                 
                   
                     T 
                     158 
                   
                 
                 
                   
                     T 
                     159 
                   
                 
                 
                   
                     T 
                     251 
                   
                 
                 
                   
                     T 
                     252 
                   
                 
                 
                   
                     T 
                     254 
                   
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
               
               
                 
                   
                     T 
                     56 
                   
                 
                 
                   
                     T 
                     55 
                   
                 
                 
                   
                     T 
                     54 
                   
                 
                 
                   
                     T 
                     53 
                   
                 
                 
                   
                     T 
                     52 
                   
                 
                 
                   
                     T 
                     155 
                   
                 
                 
                   
                     T 
                     154 
                   
                 
                 
                   
                     T 
                     153 
                   
                 
                 
                   
                     T 
                     152 
                   
                 
                 
                   
                     T 
                     60 
                   
                 
                 
                   
                     T 
                     59 
                   
                 
                 
                   
                     T 
                     57 
                   
                 
                 
                   
                     T 
                     60 
                   
                 
                 
                   
                     T 
                     154 
                   
                 
                 
                   
                     T 
                     156 
                   
                 
                 
                   
                     T 
                     158 
                   
                 
                 
                   I 
                 
               
             
              
           
         
       
       
         
           
             and 
             ⁢ 
             
                 
             
             ⁢ 
             
               M 
               2 
             
             ⁢ 
             
                 
             
             ⁢ 
             is 
           
         
       
       
         
           
              
             
               
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   
                     T 
                     2 
                   
                 
                 
                   
                     T 
                     49 
                   
                 
                 
                   
                     T 
                     50 
                   
                 
                 
                   
                     T 
                     51 
                   
                 
               
               
                 
                   
                     T 
                     56 
                   
                 
                 
                   
                     T 
                     55 
                   
                 
                 
                   
                     T 
                     54 
                   
                 
                 
                   
                     T 
                     53 
                   
                 
                 
                   
                     T 
                     52 
                   
                 
                 
                   
                     T 
                     155 
                   
                 
                 
                   
                     T 
                     154 
                   
                 
                 
                   
                     T 
                     153 
                   
                 
                 
                   
                     T 
                     152 
                   
                 
                 
                   
                     T 
                     60 
                   
                 
                 
                   
                     T 
                     59 
                   
                 
                 
                   
                     T 
                     57 
                   
                 
                 
                   
                     T 
                     60 
                   
                 
                 
                   
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                     154 
                   
                 
                 
                   
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                     156 
                   
                 
                 
                   
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                     158 
                   
                 
               
               
                 
                   I 
                 
                 
                   T 
                 
                 
                   
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                     2 
                   
                 
                 
                   
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                     3 
                   
                 
                 
                   
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                     4 
                   
                 
                 
                   
                     T 
                     156 
                   
                 
                 
                   
                     T 
                     157 
                   
                 
                 
                   
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                     158 
                   
                 
                 
                   
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                     159 
                   
                 
                 
                   
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                     251 
                   
                 
                 
                   
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                     252 
                   
                 
                 
                   
                     T 
                     254 
                   
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
                 
                   I 
                 
               
             
              
           
         
       
     
     I is an 8×8 identity matrix, I 24  is a 24×24 identity matrix. Note that the last two rows of M 2  are a permutation of the last two rows of the first 16 columns of M 1 . This property may be used for modular implementation in syndrome generation and in syndrome decoding. 
     In exemplary embodiments, the first 33 symbol positions of an ECC word are designated as the data symbols and last 3 symbol positions are designated as the check symbols. The H-matrix can also be represented as H=[H 1 , H 2 , . . . , H 36 ], where each H i  is a 24×8 binary matrix and will be call a symbol column. Note that each symbol column consists of three 8×8 sub-matrices, one of which is an identity matrix. This property is designed so that a symbol error pattern can be easily identified in syndrome decoding. All 8 bits in the same symbol are assumed to be stored in the same memory array chip. 
     Being an SSC-DSD code, the H-matrix described previously has the following properties: (1) the column vectors of each symbol column are linearly independent; (2) the dimension of the space spanned by the column vectors of any two symbol columns is 16; and (3) the dimension of the space spanned by the column vectors of any three symbol columns is 24. 
     If the number of data bits k is less than 264, the ECC can be shortened by deleting some binary columns from the data portion of the H-matrix. For example, if the number of data bits is k=256, the last symbol column (consisting of 8 binary columns) of M 1  may be deleted from the H-matrix to form a (280, 256) SSC-DSD code. If the number of data bits is k=260, bit positions 3, 4, 7, 8 of the last symbol column of M 1  may be deleted from the H-matrix to form a (284, 260) SSC-DSD code. 
     When an ECC encoded data W is received, the data is checked to see if the syndrome, S=H·W t , is an all zeros vector. If S is an all zeros vector, it is assumed that there is no error in W. Otherwise, S is decoded to determine if the errors in W is correctable and to determine the locations of errors. In exemplary embodiments described herein, a syndrome S is divided into 3 components S 1 , S 2 , and S 3  so that S=(S 1 , S 2 , S 3 ). Each of the components is an 8-bit binary vector and may be considered an element of GF(2 8 ). 
     In the example below, E i  is the error pattern at symbol position i and L i  is the symbol error indicator for symbol i. The value of L i  is one if and only if symbol i is in error. In addition,  V  is an encoded ECC word that satisfies the equation H· V   t =0 mod 2, and W= V + E (mod 2) is a corrupted word of  V  by the error vector  E =(E 1 , E 2 , . . . , E 36 ). Once the error vector  E  is determined from decoding the syndrome, the original vector  V  is recovered by setting  V =W+ E (mod 2). Thus, from the equation H· V   t 0 mod 2, the syndrome S is:
 
 S=H·W   t   =H ·(   V     t   + E     t )= H· E     t  mod 2
 
     Furthermore, if the error vector consists of a single symbol error at position i then:
 
 S=H   i   ·E   i   t  mod 2
 
     For example, if i=13, the syndrome is: 
     
       
         
           
             S 
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           T 
                           2 
                         
                       
                     
                     
                       
                         I 
                       
                     
                     
                       
                         
                           T 
                           60 
                         
                       
                     
                   
                   ] 
                 
                 · 
                 
                   E 
                   i 
                   t 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               mod 
               ⁢ 
               
                   
               
               ⁢ 
               2 
             
           
         
       
     
     In this case, S 1 =T 2 ·E i   t , S 2 =E i   t , S 3 =T 60 ·E i   t . Thus, if S 1 =T 2 ·S 2  and S 3 =T 60 ·S 2 , we set L 13 =1, and E 13 =S 2   t . All symbol error indicators can be generated in a similar way. If the syndrome is non-zero and none of the symbol indicators generated in this fashion turns on, then the UE signal is turned on. This method of syndrome decoding is a known state-of-the-art. 
     Under the assumption that the standard ECC decoder has raised an UE flag, then it is known that more than one error has occurred. The algorithm below assumes that exactly two errors have occurred, the first of which has known location (e.g., is a persistent CE). Throughout the example below, the index i identifies the known location of the first error (i.e., the CE) and the index j identifies the unknown location of a second error (i.e., the UE). Regardless of the value of i and j, it is always true that the values for the syndromes are as follows:
 
 S   1   =H   1,i   E   i   +H   1,j   E   j ;  (1)
 
 S   2   =H   2,i   E   i   +H   2,j   E   j ;  (2) and
 
 S   3   =H   3,i   E   i   +H   3,j   E   j ;  (3)
 
where Ei and Ej are the error magnitudes and they are (by assumption) nonzero. The ratios H 2,k /H 1,k , H 3,k /H 2,k  and H 3,k /H 1,k  for k in {1, . . . , 33} are key to the decoding feature under discussion. The following tables show them explicitly: for M 1  they are:
 
                                                                                                    H 3,     T 56     T 54     T 52     T 50     T 48     T 254     T 252     T 250     T 248     T 64     T 62     T 58     T 60     T 154     T 156     T 158     I       H 2,     I   T   T 2     T 3     T 4     T 156     T 157     T 158     T 159     T 251     T 252     T 254     T 253     T 206     T 205     T 204     T 28         H 3,     T 56     T 55     T 54     T 53     T 52     T 155     T 154     T 153     T 152     T 60     T 59     T 57     T 58     T 105     T 106     T 107     T 28                      
and for M 2  they are:
 
                                                                                                H 3,k /H 2,k     T 199     T 201     T 203     T 205     T 207     T 1     T 3     T 5     T 7     T 191     T 193     T 197     T 195     T 101     T 99     T 97         H 2,k /H 1,k     T 56     T 55     T 54     T 53     T 52     T 155     T 154     T 153     T 152     T 60     T 59     T 57     T 58     T 105     T 106     T 107         H 3,k /H 1,k     I   T   T 2     T 3     T 4     T 156     T 157     T 158     T 159     T 251     T 252     T 254     T 253     T 206     T 205     T 204                      
The primary property of interest is that if r and s are in {1, . . . , 33} then:
         H 3,r /H 2,r  is different from H 3,s /H 2,s      H 2,r /H 1,r  is different from H 2,s /H 1,s      H 3,r /H 1,r  is different from H 3,s /H 1,s .
 
This can be seen by direct examination of the tables shown before.
       
     Case 1. The first error has known location i in {34, 35, 36}. It is assumed that i=34. Then we know that H i,2 =H i,3 =0, and therefore formulas (1, 2, 3) translate to:
 
 S   1   =E   i   +H   1,j   E   j ;  (4)
 
S 2 =H 2,j E j ; and  (5)
 
S 3 =H 3,j E j .  (6)
 
If j=35 then S 3 =0 and S 2 =E j  and if j=36 then S 2 =0 and S 3 =E j . On the other hand, if j is in {1, . . . , 33} then neither S 2  nor S 3  can be equal to zero (because H 2,j  and H 3,j  are both nonzero). Multiplying S 2  by H 3,j /H 2,j  results in S 3 . Furthermore, for any other k in {1, . . . , 33} but not equal to j, H 3,k /H 2,k  is different from H 3,j /H 2,j  (this was observed at the beginning of this section). Therefore the following circuit depicted in  FIG. 10  would find j in the particular situation where i=34 (in the circuit, all of the output signals Lx are equal to 1 except for Lj=0 and the equality (“==”) circuit gives 0 if true and 1 otherwise). Upon finding the value of j, one must compute the error magnitudes Ei, Ej. Circuits for computing these can be found in  FIG. 11 . Similar schematics can be deduced for the cases when i=35 and i=36.
 
     Case 2. The first error has known location i in {1, . . . , 33} 
     Dividing by H 1,i , H 2,i  and H 3,i  (which are nonzero by assumption) results in:
 
 S   1   /H   1,i   =E   i +( H   1,j   /H   1,i ) E   j ;  (7)
 
 S   2   /H   2,i   =E   i +( H   2,j   /H   2,i ) E   j ; and  (8)
 
 S   3   /H   3,i   =E   i +( H   3,j   /H   3,i ) E   j .  (9)
 
The addition operation in this field is simply an XOR operation and thus addition and subtraction are identical operations. By choosing two out of three in (7, 8, 9) (a total of three possibilities) and adding the chosen equations results in:
 
 A   1+2   =S   1   /H   1,i   +S   2   /H   2,i =( H   1,j   /H   1,i   +H   2,j   /H   2,i ) E   j ;  (10)
 
 A   2+3   S   2   /H   2,i   +S   3   /H   3,i =( H   2,j   /H   2,i   +H   3,j   /H   3,i ) E   j ; and  (11)
 
 A   1+3   =S   1   /H   1,i   +S   3   /H   3,i =( H   1,j   /H   1,i   +H   3,j   /H   3,i ) E   j .  (12)
 
If j is in {34, 35, 36} then exactly one of the above will be zero. Furthermore, if j is in {1, . . . , 33}, then none of A 1+2 , A 2+3 , A 1+3  can be equal to zero; this can be seen from the properties that we derived of the check matrix H at the beginning of this discussion. Assume that j=34. Then H 2,j =H 3,j =0 and therefore A 2+3 =0. Moreover, A 1+2 =A 1+3 =E j /H 1,j . A similar analysis for the cases j=35 and j=36 yields the following table.
 
     
       
         
           
               
               
               
               
               
             
               
                   
                   
               
               
                   
                 i in {1, . . . , 33} 
                 A 1+2   
                 A 2+3   
                 A 1+3   
               
               
                   
                   
               
             
            
               
                   
                 j = 34 
                 E j /H 1,i   
                 0 
                 E j /H 1,i   
               
               
                   
                 j = 35 
                 E j /H 2,i   
                 E j /H 2,i   
                 0 
               
               
                   
                 j = 36 
                 0 
                 E j /H 3,i   
                 E j /H 3,i   
               
               
                   
                   
               
            
           
         
       
     
     Note that if j is in {1, . . . , 33} and A 1+2 =0, then H 1,j /H 1,i =H 2,j /H 2,i . Manipulating this, results in H 2,j /H 1,j =H 2,i /H 2,j  and since by assumption i is different from j, from the properties of H described at the beginning of this section results in a contradiction. Similarly if A 2+3 =0 then H 3,j /H 2,j =H 3,i /H 2,i  and if A 1+3 =0 then H 3,j /H 1,j =H 3,i /H 1,i , results in similar contradictions. 
     The discussion above implies that the existence of one zero in (A 1+2 , A 2+3 , A 1+3 ) can be used in order to discriminate between the cases where j is in {1, . . . , 33} and j is in {34, 35, 36}. Furthermore in the case where j is in {34, 35, 36}, the table above immediately gives a technique to find j, E j  and through (1, 2, 3), E i . 
     It remains how to decode when j is in {1, . . . , 33}. By direct examination of the parity check matrix H it can be deduced that for every index k in the set {1, . . . , 33}, the parity check matrix satisfies: H 2,k  H 3,k =T 56  (H 1,k ) 2 . Forming the ratio A 2+3 /A 1+2 , using the above to replace H 3,i  and H 3,j  and simplifying the resulting expression results in: 
                       A     2   +   3       /     A     1   +   2         =     1   +           H     1   ,   j       /     H     2   ,   j             H     1   ,   i       /     H     2   ,   i           .               (   13   )               
The ratio H 1,i /H 2,i  is known and the left hand side of the equation above is known, thus it may be solved for H 1,j /H 2,j  via:
   H   1,j   /H   2,j =( A   2+3   /A   1+2 +1)( H   1,j   /H   2,i ) 
     In order to find the error location j, the unique values of j much be identified such that the equation above is satisfied. The uniqueness follows from the properties of H discussed previously. Having found the error location j, the error magnitudes are found by the following: 
     
       
         
           
             
               
                 
                   
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                             i 
                           
                         
                       
                       
                         
                           
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                   = 
                   
                     
                       
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                                 1 
                                 , 
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                             ⁢ 
                             
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                           + 
                           
                             
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                             ⁢ 
                             
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                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
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                                   , 
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                                   , 
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                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               S 
                               1 
                             
                           
                         
                         
                           
                             
                               S 
                               2 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
       FIGS. 12 and 13  depict the calculation of A 1+2 , A 2+3  and A 1+3  together with the error location algorithm as a circuit schematic. 
     The above demonstrates one set of computations that can be carried out by exemplary embodiments in order to correct two errors, the first of which is in a known location. For implementation considerations, it desirable to avoid the task of implementing a Galois field division circuit that will divide a Galois field element by an arbitrary element. 
     The task of forming A 1+2  and A 2+3  can be accomplished by computing in parallel for every k in {1, . . . , 33} the three quantities S 1 /H 1,k , S 2 /H 2,k  and S 3 /H 3,k , and then selecting the correct computation using a multiplexer whose input is i, the known location for the error. Since the elements H 1,k , H 2,k  and H 3,k  are known in advance, their inverses can be pre-computed and the task above can be turned into a multiplication. 
     In order to avoid computing A 2+3 /A 1+2  note that it must be true that:
 
( H   1,j   /H   2,j ) A   1+2 =( A   1+2   +A   2+3 )( H   1,j   /H   2,i ).
 
To find the unique value of j for which the above holds, the products (H 1k /H 2k )A 1+2  are formed in parallel for every k in {1, . . . , 33} and a check is made for equality with the right hand side. The factor (H 1,i /H 2,i ) is obtained by using a multiplexer with hard-wired inputs and selector i.
 
     One Galois field inversion still remains for the computation of the error magnitudes in (18) (after the computation of this inversion, the result can be shared to compute both E i  and E j ). In exemplary embodiments, the denominator H 1,i  H 2,j +H 1,j  H 2 , is computed and then the result is inverted via a memory with 256 8-bit entries. 
     Exemplary embodiments include a computing system with a processor(s) and an I/O unit(s) (e.g., requestors) interconnected to a memory system that contains a memory controller and memory devices. In exemplary embodiments, the memory system includes a processor or memory controller interfaced to a set of hub devices (also referred to as “hub chips”). The hub devices connect and interface to the memory devices. In exemplary embodiments the computer memory system includes a physical memory array with a plurality of memory devices for storing data and instructions. These memory devices may be connected directly to the memory controller and/or indirectly coupled to the memory controller through hub devices. In exemplary embodiments, the hub-based computer memory system has memory devices attached to a communication hub device that is connected to a memory control device (e.g., a memory controller). Also in exemplary embodiments, the hub device is located on a memory module (e.g., a single substrate or physical device) that includes two or more hub devices that are cascaded interconnected to each other (and possibly to another hub device located on another memory module) via the memory bus. 
     Hub devices may be connected to the memory controller through a multi-drop or point-to-point bus structure (which may further include a cascade connection to one or more additional hub devices). Memory access requests are transmitted by the memory controller through the bus structure (e.g., the memory bus) to the selected hub(s). In response to receiving the memory access requests, the hub device translates the memory access requests to control the memory devices to store write data from the hub device or to provide read data to the hub device. Read data is encoded into one or more communication packet(s) and transmitted through the memory bus(ses) to the memory controller. 
     In alternate exemplary embodiments, the memory controller(s) may be integrated together with one or more processor chips and supporting logic, packaged in a discrete chip (commonly called a “northbridge” chip), included in a multi-chip carrier with the one or more processors and/or supporting logic, or packaged in various alternative forms that best match the application/environment. Any of these solutions may or may not employ one or more narrow/high speed links to connect to one or more hub chips and/or memory devices. 
     The memory modules may be implemented by a variety of technology including a DIMM, a single in-line memory module (SIMM) and/or other memory module or card structures. In general, a DIMM refers to a small circuit board which is comprised primarily of random access memory (RAM) integrated circuits or die on one or both sides with signal and/or power pins on both sides of the board. This can be contrasted to a SIMM which is a small circuit board or substrate composed primarily of RAM integrated circuits or die on one or both sides and single row of pins along one long edge. The DIMM depicted in  FIG. 1  includes 168 pins in the exemplary embodiment, whereas subsequent DIMMs have been constructed with pincounts ranging from 100 pins to over 300 pins. In exemplary embodiments described herein, memory modules may include two or more hub devices. 
     In exemplary embodiments, the memory bus is constructed using multi-drop connections to hub devices on the memory modules and/or using point-to-point connections. The downstream portion of the controller interface (or memory bus), referred to as the downstream bus, may include command, address, data and other operational, initialization or status information being sent to the hub devices on the memory modules. Each hub device may simply forward the information to the subsequent hub device(s) via bypass circuitry; receive, interpret and re-drive the information if it is determined to be targeting a downstream hub device; re-drive some or all of the information without first interpreting the information to determine the intended recipient; or perform a subset or combination of these options. 
     The upstream portion of the memory bus, referred to as the upstream bus, returns requested read data and/or error, status or other operational information, and this information may be forwarded to the subsequent hub devices via bypass circuitry; be received, interpreted and re-driven if it is determined to be targeting an upstream hub device and/or memory controller in the processor complex; be re-driven in part or in total without first interpreting the information to determine the intended recipient; or perform a subset or combination of these options. 
     In alternate exemplary embodiments, the point-to-point bus includes a switch or bypass mechanism which results in the bus information being directed to one of two or more possible hub devices during downstream communication (communication passing from the memory controller to a hub device on a memory module), as well as directing upstream information (communication from a hub device on a memory module to the memory controller), often by way of one or more upstream hub devices. Further embodiments include the use of continuity modules, such as those recognized in the art, which, for example, can be placed between the memory controller and a first populated hub device (i.e., a hub device that is in communication with one or more memory devices), in a cascade interconnect memory system, such that any intermediate hub device positions between the memory controller and the first populated hub device include a means by which information passing between the memory controller and the first populated hub device can be received even if the one or more intermediate hub device position(s) do not include a hub device. The continuity module(s) may be installed in any module position(s), subject to any bus restrictions, including the first position (closest to the main memory controller, the last position (prior to any included termination) or any intermediate position(s). The use of continuity modules may be especially beneficial in a multi-module cascade interconnect bus structure, where an intermediate hub device on a memory module is removed and replaced by a continuity module, such that the system continues to operate after the removal of the intermediate hub device. In more common embodiments, the continuity module(s) would include either interconnect wires to transfer all required signals from the input(s) to the corresponding output(s), or be re-driven through a repeater device. The continuity module(s) might further include a non-volatile storage device (such as an EEPROM), but would not include main memory storage devices. 
     In exemplary embodiments, the memory system includes one or more hub devices on one or more memory modules connected to the memory controller via a cascade interconnect memory bus, however other memory structures may be implemented such as a point-to-point bus, a multi-drop memory bus or a shared bus. Depending on the signaling methods used, the target operating frequencies, space, power, cost, and other constraints, various alternate bus structures may be considered. A point-to-point bus may provide the optimal performance in systems produced with electrical interconnections, due to the reduced signal degradation that may occur as compared to bus structures having branched signal lines, switch devices, or stubs. However, when used in systems requiring communication with multiple devices or subsystems, this method will often result in significant added component cost and increased system power, and may reduce the potential memory density due to the need for intermediate buffering and/or re-drive. 
     Although not shown in the Figures, the memory modules or hub devices may also include a separate bus, such as a ‘presence detect’ bus, an I2C bus and/or an SMBus which is used for one or more purposes including the determination of the hub device an/or memory module attributes (generally after power-up), the reporting of fault or status information to the system, the configuration of the hub device(s) and/or memory subsystem(s) after power-up or during normal operation or other purposes. Depending on the bus characteristics, this bus might also provide a means by which the valid completion of operations could be reported by the hub devices and/or memory module(s) to the memory controller(s), or the identification of failures occurring during the execution of the main memory controller requests. 
     Performances similar to those obtained from point-to-point bus structures can be obtained by adding switch devices. These and other solutions offer increased memory packaging density at lower power, while retaining many of the characteristics of a point-to-point bus. Multi-drop busses provide an alternate solution, albeit often limited to a lower operating frequency, but at a cost/performance point that may be advantageous for many applications. Optical bus solutions permit significantly increased frequency and bandwidth potential, either in point-to-point or multi-drop applications, but may incur cost and space impacts. 
     As used herein the term “buffer” or “buffer device” refers to a temporary storage unit (as in a computer), especially one that accepts information at one rate and delivers it another. In exemplary embodiments, a buffer is an electronic device that provides compatibility between two signals (e.g., changing voltage levels or current capability). The term “hub” is sometimes used interchangeably with the term “buffer.” A hub is a device containing multiple ports that is connected to several other devices. A port is a portion of an interface that serves a congruent I/O functionality (e.g., a port may be utilized for sending and receiving data, address, and control information over one of the point-to-point links, or busses). A hub may be a central device that connects several systems, subsystems, or networks together. A passive hub may simply forward messages, while an active hub, or repeater, amplifies and refreshes the stream of data which otherwise would deteriorate over a distance. The term hub device, as used herein, refers to a hub chip that includes logic (hardware and/or software) for performing memory functions. 
     Also as used herein, the term “bus” refers to one of the sets of conductors (e.g., wires, and printed circuit board traces or connections in an integrated circuit) connecting two or more functional units in a computer. The data bus, address bus and control signals, despite their names, constitute a single bus since each are often useless without the others. A bus may include a plurality of signal lines, each signal line having two or more connection points, that form a main transmission path that electrically connects two or more transceivers, transmitters and/or receivers. The term “bus” is contrasted with the term “channel” which is often used to describe the function of a “port” as related to a memory controller in a memory system, and which may include one or more busses or sets of busses. The term “channel” as used herein refers to a port on a memory controller. Note that this term is often used in conjunction with I/O or other peripheral equipment, however the term channel has been adopted by some to describe the interface between a processor or memory controller and one of one or more memory subsystem(s). 
     Further, as used herein, the term “daisy chain” refers to a bus wiring structure in which, for example, device A is wired to device B, device B is wired to device C, etc. The last device is typically wired to a resistor or terminator. All devices may receive identical signals or, in contrast to a simple bus, each device may modify one or more signals before passing them on. A “cascade” or cascade interconnect’ as used herein refers to a succession of stages or units or a collection of interconnected networking devices, typically hubs, in which the hubs operate as a logical repeater, further permitting merging data to be concentrated into the existing data stream. Also as used herein, the term “point-to-point” bus and/or link refers to one or a plurality of signal lines that may each include one or more terminators. In a point-to-point bus and/or link, each signal line has two transceiver connection points, with each transceiver connection point coupled to transmitter circuitry, receiver circuitry or transceiver circuitry. A signal line refers to one or more electrical conductors or optical carriers, generally configured as a single carrier or as two or more carriers, in a twisted, parallel, or concentric arrangement, used to transport at least one logical signal. 
     Memory devices are generally defined as integrated circuits that are composed primarily of memory (storage) cells, such as DRAMs (Dynamic Random Access Memories), SRAMs (Static Random Access Memories), FeRAMs (Ferro-Electric RAMs), MRAMs (Magnetic Random Access Memories), Flash Memory and other forms of random access and related memories that store information in the form of electrical, optical, magnetic, biological or other means. Dynamic memory device types may include asynchronous memory devices such as FPM DRAMs (Fast Page Mode Dynamic Random Access Memories), EDO (Extended Data Out) DRAMs, BEDO (Burst EDO) DRAMs, SDR (Single Data Rate) Synchronous DRAMs, DDR (Double Data Rate) Synchronous DRAMs or any of the expected follow-on devices such as DDR2, DDR3, DDR4 and related technologies such as Graphics RAMs, Video RAMs, LP RAM (Low Power DRAMs) which are often based on the fundamental functions, features and/or interfaces found on related DRAMs. 
     Memory devices may be utilized in the form of chips (die) and/or single or multi-chip packages of various types and configurations. In multi-chip packages, the memory devices may be packaged with other device types such as other memory devices, logic chips, analog devices and programmable devices, and may also include passive devices such as resistors, capacitors and inductors. These packages may include an integrated heat sink or other cooling enhancements, which may be further attached to the immediate carrier or another nearby carrier or heat removal system. 
     Module support devices (such as buffers, hubs, hub logic chips, registers, PLL&#39;s, DLL&#39;s, non-volatile memory, etc) may be comprised of multiple separate chips and/or components, may be combined as multiple separate chips onto one or more substrates, may be combined onto a single package or even integrated onto a single device—based on technology, power, space, cost and other tradeoffs. In addition, one or more of the various passive devices such as resistors, capacitors may be integrated into the support chip packages, or into the substrate, board or raw card itself, based on technology, power, space, cost and other tradeoffs. These packages may include an integrated heat sink or other cooling enhancements, which may be further attached to the immediate carrier or another nearby carrier or heat removal system. 
     Memory devices, hubs, buffers, registers, clock devices, passives and other memory support devices and/or components may be attached to the memory subsystem and/or hub device via various methods including solder interconnects, conductive adhesives, socket structures, pressure contacts and other methods which enable communication between the two or more devices via electrical, optical or alternate means. 
     The one or more memory modules (or memory subsystems) and/or hub devices may be connected to the memory system, processor complex, computer system or other system environment via one or more methods such as soldered interconnects, connectors, pressure contacts, conductive adhesives, optical interconnects and other communication and power delivery methods. Connector systems may include mating connectors (male/female), conductive contacts and/or pins on one carrier mating with a male or female connector, optical connections, pressure contacts (often in conjunction with a retaining mechanism) and/or one or more of various other communication and power delivery methods. The interconnection(s) may be disposed along one or more edges of the memory assembly and/or placed a distance from an edge of the memory subsystem depending on such application requirements as ease-of-upgrade/repair, available space/volume, heat transfer, component size and shape and other related physical, electrical, optical, visual/physical access, etc. 
     As used herein, the term memory subsystem refers to, but is not limited to: one or more memory devices; one or more memory devices and associated interface and/or timing/control circuitry; and/or one or more memory devices in conjunction with a memory buffer, hub device, and/or switch. The term memory subsystem may also refer to one or more memory devices, in addition to any associated interface and/or timing/control circuitry and/or a memory buffer, hub device or switch, assembled into a substrate, a card, a module or related assembly, which may also include a connector or similar means of electrically attaching the memory subsystem with other circuitry. The memory modules described herein may also be referred to as memory subsystems because they include one or more memory devices and hub devices 
     Additional functions that may reside local to the memory subsystem and/or hub device include write and/or read buffers, one or more levels of memory cache, local pre-fetch logic, data encryption/decryption, compression/decompression, protocol translation, command prioritization logic, voltage and/or level translation, error detection and/or correction circuitry, data scrubbing, local power management circuitry and/or reporting, operational and/or status registers, initialization circuitry, performance monitoring and/or control, one or more co-processors, search engine(s) and other functions that may have previously resided in other memory subsystems. By placing a function local to the memory subsystem, added performance may be obtained as related to the specific function, often while making use of unused circuits within the subsystem. 
     Memory subsystem support device(s) may be directly attached to the same substrate or assembly onto which the memory device(s) are attached, or may be mounted to a separate interposer or substrate also produced using one or more of various plastic, silicon, ceramic or other materials which include electrical, optical or other communication paths to functionally interconnect the support device(s) to the memory device(s) and/or to other elements of the memory or computer system. 
     Information transfers (e.g., packets) along a bus, channel, link or other naming convention applied to an interconnection method may be completed using one or more of many signaling options. These signaling options may include such methods as single-ended, differential, optical or other approaches, with electrical signaling further including such methods as voltage or current signaling using either single or multi-level approaches. Signals may also be modulated using such methods as time or frequency, non-return to zero, phase shift keying, amplitude modulation and others. Voltage levels are expected to continue to decrease, with 1.5V, 1.2V, 1V and lower signal voltages expected consistent with (but often independent of) the reduced power supply voltages required for the operation of the associated integrated circuits themselves. 
     One or more clocking methods may be utilized within the memory subsystem and the memory system itself, including global clocking, source-synchronous clocking, encoded clocking or combinations of these and other methods. The clock signaling may be identical to that of the signal lines themselves, or may utilize one of the listed or alternate methods that is more conducive to the planned clock frequency(ies), and the number of clocks planned within the various subsystems. A single clock may be associated with all communication to and from the memory, as well as all clocked functions within the memory subsystem, or multiple clocks may be sourced using one or more methods such as those described earlier. When multiple clocks are used, the functions within the memory subsystem may be associated with a clock that is uniquely sourced to the subsystem, or may be based on a clock that is derived from the clock related to the information being transferred to and from the memory subsystem (such as that associated with an encoded clock). Alternately, a unique clock may be used for the information transferred to the memory subsystem, and a separate clock for information sourced from one (or more) of the memory subsystems. The clocks themselves may operate at the same or frequency multiple of the communication or functional frequency, and may be edge-aligned, center-aligned or placed in an alternate timing position relative to the data, command or address information. 
     Information passing to the memory subsystem(s) will generally be composed of address, command and data, as well as other signals generally associated with requesting or reporting status or error conditions, resetting the memory, completing memory or logic initialization and other functional, configuration or related information. Information passing from the memory subsystem(s) may include any or all of the information passing to the memory subsystem(s), however generally will not include address and command information. This information may be communicated using communication methods that may be consistent with normal memory device interface specifications (generally parallel in nature), the information may be encoded into a ‘packet’ structure, which may be consistent with future memory interfaces or simply developed to increase communication bandwidth and/or enable the subsystem to operate independently of the memory technology by converting the received information into the format required by the receiving device(s). 
     Initialization of the memory subsystem may be completed via one or more methods, based on the available interface busses, the desired initialization speed, available space, cost/complexity objectives, subsystem interconnect structures, the use of alternate processors (such as a service processor) which may be used for this and other purposes, etc. In one embodiment, the high speed bus may be used to complete the initialization of the memory subsystem(s), generally by first completing a training process to establish reliable communication, then by interrogation of the attribute or ‘presence detect’ data associated the various components and/or characteristics associated with that subsystem, and ultimately by programming the appropriate devices with information associated with the intended operation within that system. In a cascaded system, communication with the first memory subsystem would generally be established, followed by subsequent (downstream) subsystems in the sequence consistent with their position along the cascade interconnect bus. 
     A second initialization method would include one in which the high speed bus is operated at one frequency during the initialization process, then at a second (and generally higher) frequency during the normal operation. In this embodiment, it may be possible to initiate communication with all of the memory subsystems on the cascade interconnect bus prior to completing the interrogation and/or programming of each subsystem, due to the increased timing margins associated with the lower frequency operation. 
     A third initialization method might include operation of the cascade interconnect bus at the normal operational frequency(ies), while increasing the number of cycles associated with each address, command and/or data transfer. In one embodiment, a packet containing all or a portion of the address, command and/or data information might be transferred in one clock cycle during normal operation, but the same amount and/or type of information might be transferred over two, three or more cycles during initialization. This initialization process would therefore be using a form of ‘slow’ commands, rather than ‘normal’ commands, and this mode might be automatically entered at some point after power-up and/or re-start by each of the subsystems and the memory controller by way of POR (power-on-reset) logic included in each of these subsystems. 
     A fourth initialization method might utilize a distinct bus, such as a presence detect bus (such as the one defined in U.S. Pat. No. 5,513,135 to Dell et al., of common assignment herewith), an I2C bus (such as defined in published JEDEC standards such as the 168 Pin DIMM family in publication 21-C revision 7R8) and/or the SMBUS, which has been widely utilized and documented in computer systems using such memory modules. This bus might be connected to one or more modules within a memory system in a daisy chain/cascade interconnect, multi-drop or alternate structure, providing an independent means of interrogating memory subsystems, programming each of the one or more memory subsystems to operate within the overall system environment, and adjusting the operational characteristics at other times during the normal system operation based on performance, thermal, configuration or other changes desired or detected in the system environment. 
     Other methods for initialization can also be used, in conjunction with or independent of those listed. The use of a separate bus, such as described in the fourth embodiment above, also offers the advantage of providing an independent means for both initialization and uses other than initialization, such as described in U.S. Pat. No. 6,381,685 to Dell et al., of common assignment herewith, including changes to the subsystem operational characteristics on-the-fly and for the reporting of and response to operational subsystem information such as utilization, temperature data, failure information or other purposes. 
     With improvements in lithography, better process controls, the use of materials with lower resistance, increased field sizes and other semiconductor processing improvements, increased device circuit density (often in conjunction with increased die sizes) will help facilitate increased function on integrated devices as well as the integration of functions previously implemented on separate devices. This integration will serve to improve overall performance of the intended function, as well as promote increased storage density, reduced power, reduced space requirements, lower cost and other manufacturer and customer benefits. This integration is a natural evolutionary process, and may result in the need for structural changes to the fundamental building blocks associated with systems. 
     The integrity of the communication path, the data storage contents and all functional operations associated with each element of a memory system or subsystem can be assured, to a high degree, with the use of one or more fault detection and/or correction methods. Further reliability enhancements may include operation re-try (to overcome intermittent faults such as those associated with the transfer of information), the use of one or more alternate or replacement communication paths to replace failing paths and/or lines, complement-re-complement techniques or alternate methods used in computer, communication and related systems. 
     The use of bus termination, on busses as simple as point-to-point links or as complex as multi-drop structures, is becoming more common consistent with increased performance demands. A wide variety of termination methods can be identified and/or considered, and include the use of such devices as resistors, capacitors, inductors or any combination thereof, with these devices connected between the signal line and a power supply voltage or ground, a termination voltage or another signal. The termination device(s) may be part of a passive or active termination structure, and may reside in one or more positions along one or more of the signal lines, and/or as part of the transmitter and/or receiving device(s). The terminator may be selected to match the impedance of the transmission line, or selected via an alternate approach to maximize the useable frequency, operating margins and related attributes within the cost, space, power and other constraints. 
     Technical effects and benefits of exemplary embodiments include the ability to identify a failing memory module in a memory system. This may lead to a reduction in materials cost because all of the memory modules in the system will not have to be replaced when only one is in need of repair. In addition, this should result in minimizing the downtime of the memory system and increasing the probability that the failure has been resolved. 
     As described above, the embodiments of the invention may be embodied in the form of computer-implemented processes and apparatuses for practicing those processes. Embodiments of the invention may also be embodied in the form of computer program code containing instructions embodied in tangible media, such as floppy diskettes, CD-ROMs, hard drives, or any other computer-readable storage medium, wherein, when the computer program code is loaded into and executed by a computer, the computer becomes an apparatus for practicing the invention. The present invention can also be embodied in the form of computer program code, for example, whether stored in a storage medium, loaded into and/or executed by a computer, or transmitted over some transmission medium, such as over electrical wiring or cabling, through fiber optics, or via electromagnetic radiation, wherein, when the computer program code is loaded into and executed by a computer, the computer becomes an apparatus for practicing the invention. When implemented on a general-purpose microprocessor, the computer program code segments configure the microprocessor to create specific logic circuits. 
     While the invention has been described with reference to exemplary embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed as the best mode contemplated for carrying out this invention, but that the invention will include all embodiments falling within the scope of the appended claims. Moreover, the use of the terms first, second, etc. do not denote any order or importance, but rather the terms first, second, etc. are used to distinguish one element from another.