# Patent

## Title
Multi-level run-length limited finite state machine with multi-penalty

## Abstract

Techniques are described for constructing maximum transition run (MTR) modulation code based upon a multi-level (ML) run-length limited (RLL) finite state machine (FSM) that implements different sets of penalties. A processor is configured to receive information from a hard disk drive (HDD) via a read channel and recover data from the HDD using MTR modulation code. A memory has computer executable instructions configured for execution by the processor to model a magnetic recording channel as a partial response channel, model a source of information to the magnetic recording channel to provide an optimized Markov source, and construct an MTR modulation code to mimic the optimized Markov source based upon an FSM having a limited transition run length and a multi-level periodic structure. The FSM provides at least two different sets of penalties in a period.

## Background
BACKGROUND
Various data processing systems have been developed including storage systems, cellular telephone systems, and radio transmission systems. In such systems, data is transferred from a sender to a receiver via a medium. For example, in a storage system, data is sent from a sender (e.g., a write function) to a receiver (e.g., a read function) via a storage medium. As information is stored and transmitted in the form of digital data, errors are introduced that, if not corrected, can corrupt the data and render the information unusable. The effectiveness of any transfer is impacted by any losses in data caused by various factors.
Consequently, error checking systems have been developed to detect and correct errors of digital data. Error checking systems may be used, for example, to process data retrieved from a magnetic hard disk drive (HDD). Each data sector of the disk drive may have different noise, jitter, and distortion characteristics or signal to noise ratios (SNR), which may be due, for example, to magnetic media defects, off-track writing, high fly height of magnetic write heads during a writing operation, large phase disturbance, and so forth. The throughput of an HDD can be affected by the number of read errors in a data sector, based in part on the SNR, and by the speed at which the read channel can recover from a read error by correcting the errors.
SUMMARY
Techniques are described for constructing maximum transition run (MTR) modulation code based upon a multi-level (ML) run-length limited (RLL) finite state machine (FSM) that implements different sets of penalties. A processor is configured to receive information from a hard disk drive (HDD) via a read channel and recover data from the HDD using MTR modulation code. A memory has computer executable instructions configured for execution by the processor to model a magnetic recording channel as a partial response channel, model a source of information to the magnetic recording channel to provide an optimized Markov source, and construct an MTR modulation code to mimic the optimized Markov source based upon an FSM having a limited transition run length and a multi-level periodic structure. The FSM provides at least two different sets of penalties in a period.
This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter.



DRAWINGS
The detailed description is described with reference to the accompanying figures. In the figures, the left-most digit(s) of a reference number identifies the figure in which the reference number first appears. The use of the same reference number in different instances in the description and the figures may indicate similar or identical items.
FIG. 1A is a block diagram illustrating a Markov source optimization for a magnetic recording channel in accordance with an example implementation of the present disclosure.
FIG. 1B is a block diagram illustrating a Markov source optimization, where data dependent autoregressive (AR) modeling is used to model a magnetic recording channel in accordance with an example implementation of the present disclosure.
FIG. 2A is a diagrammatic illustration of a finite state machine (FSM) for maximum transition run (MTR) code where j=4 (mtr4 code) in accordance with an example implementation of the present disclosure.
FIG. 2B is a diagrammatic illustration of a single level mtr4 FSM with two time ticks in accordance with an example implementation of the present disclosure.
FIG. 2C is a diagrammatic illustration of an FSM with a multi-level (ML) periodic structure, where a timeline is included in accordance with an example implementation of the present disclosure.
FIG. 3 is a diagrammatic illustration of a run-length limited (RLL) FSM with an ML periodic structure implementing two sets of penalties, where a timeline is included in accordance with an example implementation of the present disclosure.
FIG. 4 is a flow diagram illustrating a method for constructing MTR modulation code, e.g., for recovering data from a hard disk drive (HDD) in accordance with example implementations of the present disclosure.
FIG. 5 is a block diagram illustrating a storage system having information divergence based data processing circuitry in accordance with an example implementation of the present disclosure.
FIG. 6 is a block diagram illustrating a data transmission system having information divergence based data processing circuitry in accordance with an example implementation of the present disclosure.



DETAILED DESCRIPTION
Techniques are described to implement a flexible multi-level structure for a finite state machine. Techniques of the present disclosure can be used to provide a multi-level penalty structure for a run-length limited finite state machine, which can generate a nearly optimal sequence for a magnetic recording channel, and can be used with various HDD signal processing techniques, including algorithms, digital signal processing (DSP), coding, read channel, and so forth.
Jitter noise is the major noise source in a magnetic recording channel and is caused by transitions in channel sequence. Transition distribution of a channel sequence generally determines the channel noise level/signature, channel information rate, and, thus, final error rate performance. In high density magnetic recording, burst errors caused by long transition runs are dominant in read channel implementations. To eliminate long transition runs, maximum transition run (MTR) codes can be used as modulation codes in a magnetic recording channel. For example, an MTR(j) code, where j refers to the length of a transition run, can be used to terminate transition runs longer than j. However, MTR coded channels also incur additional channel density penalty and performance degradation due to the code rate loss. For example, MTR (j=3) code (mtr3 code) does not offer an information rate greater than at least approximately 0.9468. In order to construct a finite state machine that can generate sequences with better transitions than mtr3 code, a Markov source can be optimized to find optimized transition distributions of channel sequence for a particular density of interest, and finite state machines (FSMs) can be constructed with multi-level structure to shape the transition distribution close to the optimized transition distribution. In this manner, FSMs can be constructed that have a wide range of capacity (code rate), e.g., ranging from at least approximately 0.94 to 0.96. In some instances, these FSMs can offer performance improvement over, for example, an mtr3 coded channel.
Channel capacity can be used as an indication of the highest code rate of error correction code (ECC) that can provide error free reception at a receiver. However, to achieve better performance given a fixed ECC code rate, it may be desirable to have a higher mutual information rate (e.g., approaching channel capacity). By optimizing the source distribution, the channel mutual information rate can be maximized to be very close to the channel capacity. A magnetic recording channel can be modeled as a partial response channel having memory. A source that provides information to the magnetic recording channel can also be modeled as having memory, e.g., during an optimization, which can be referred to as a Markov source optimization. A Markov source optimization is an iterative procedure, which is considered to converge when the information rate stops increasing, e.g., as illustrated in FIG. 1A, where a magnetic recording channel module 100A is supplied with an initial uniform distribution of data 102 via a Markov source module 104, the data is then supplied from the magnetic recording channel module 100A to an a posteriori probability (APP) detector module 106 and subsequently to an information rate computation module 108, and where a loop is provided from the APP detector module 106 back to the Markov source module 104 via a noisy adjacency matrix computation module 110 (connected to the APP detector module 106) and a distribution adjustment module 112 (connected to the noisy adjacency matrix computation module 110 and the Markov source module 104). In some instances, the Markov source module 104 can be implemented using pseudo code, and the performance of the pseudo code can be verified by simulation.
The magnetic recording channel (MRC) can be subject to complicated noise factors (e.g., both linear and nonlinear), which may not be captured by the APP detector module 106 in the optimization loop illustrated in FIG. 1A. Thus, the mutual information rate may not be accurately estimated during optimization, and/or a particular iteration may not be stable (e.g., resulting in an ill-conditioned distribution). In order to provide a Markov source module 104 that is optimized accurately and/or stably, the APP detector module 106 can be matched to the channel exactly, or at least substantially exactly, where data dependent noise is completely, or at least substantially completely, whitened. In one particular instance, e.g., as illustrated in FIG. 1B, data dependent autoregressive (AR) modeling can be used to model the MRC (e.g., using a data dependent AR modeling module 100B). However, this modeling technique is provided by way of example only and is not meant to be restrictive of the present disclosure. Thus, in other implementations, other various modeling techniques can be used. With the data dependent AR modeling implementation illustrated in FIG. 1B, the data dependent noise whitening filters in the APP detector module 106 are exactly the inverse of the AR model illustrated by AR modeling module 100B.
A code can be designed to mimic an optimized Markov source, where the optimized Markov source is determined as described above with reference to FIGS. 1A and 1B. However, providing such a code may be difficult when 0.0x decibel (dB) gain over MTR codes is desired, and where perfect matching on the distribution is preferred. Further high rate codes can be desirable (e.g., according to the capacity of the Markov source). However, if the source distribution is not matched directly, codes can be designed to match the transition run length distributions of the optimized Markov sources. For instance, a modulation code can be constructed based upon a finite state machine (FSM), which can be used to define all properties of a sequence, and can achieve a desired error rate performance. It should be noted that a code implementation based upon an FSM may use an enumerative technique, state splitting, and so on. In a particular instance, an FSM can be used with limited transition run length (although the optimized Markov source does not have a limited transition run length). Accordingly, code based upon the FSM in this implementation can be MTR code. In a particular instance, the code rate can range from at least approximately 0.9468 to 0.9614. For example, since mtr3 code has a capacity of at least approximately 0.9468 and there may not be space to suppress the distribution, mtr4, mtr5, and code having longer limits can also be considered. FIG. 2A illustrates a basic representation of an FSM for mtr4 code.
Referring now to FIG. 2B, a multi-level (ML) periodic structure can be used to form an FSM. This technique can be used to suppress the long transitions inherent in, for example, a simple mtr4 FSM (e.g., the FSM as illustrated in FIG. 2A). For the purposes of the present disclosure, the term “periodic” is used to refer to an FSM that has an output granularity greater than one (1) binary digit (bit), such as the granularity of a block (e.g., where the block length is the period). Referring to FIG. 2C, an example FSM is illustrated with an accompanying timeline. To suppress the transitions, an ML structure is used. Given a fixed number of levels, where the sequence starts from the lowest level, the sequence flows upward when transitions occur that are to be suppressed. In this manner, “penalties” are added on the transition patterns in the FSM. It should be noted that states in the top level have no upwardly extending branches. In operation, as the last time increment (e.g., tick) of the period is encountered, the constraint can be loosened by lowering the group of connections by several levels, which can be referred to as a “drop.” In the accompanying FIG. 2C, the level of the FSM is equal to sixteen (16), the penalty is equal to [0, 1, 2, 3] upward levels for a single transition, and the drop is equal to nine (9). In this figure, the last transition is illustrated in double, triple, and four transition runs.
In implementations, the ML-FSM can have one or more of the following characteristics. The capacity of the FSM can be greater than the code rate needed for a particular configuration. The transition distribution of the FSM can approach the optimized transition distribution. Penalties can be time-invariant. In some instances, an exhaustive search can be used to produce an ML-FSM structure. For instance, the following search algorithm can be used to generate an ML-FSM. As described herein, a search algorithm is provided using pseudo code, where [p1, p2, p3, p4] represents penalties for the first, second, third, and fourth transitions, respectively; P_limit is a predefined limit for the penalties; and L_limit is the limit set for a particular level. Then, for a particular period,












N_P = [p1, p2, p3, p4] * [1, P_limit, P_limit{circumflex over ( )}2, P_limit{circumflex over ( )}3]T;


for (level = 2; level <= L_limit; level++) {


  for (drop = level − 1; drop > 0; drop−−) {


   for (p = 1; p <= N_P; p++) {


     convert p to [p1, p2, p3, p4] then construct the FSM


     compute the capacity of the FSM, Cfsm, and the probabilities


     for dc, 1, 2, 3 and 4 transition runs, i.e., the distribution


     if Cfsm > k/n and the distribution is close to the optimized


     one, then this setting [level, drop, p1, p2, p3, p4] is saved to


     a list for later processing


   }


  }


}









After executing the preceding algorithm, a list of possible FSMs is available. One or more FSMs can be selected according to, for instance, FSMs that provide sufficient large capacity, transitions that are close to an optimized transition, and so forth. For the purposes of the present disclosure, the term “close” can be used to refer to transitions with probabilities in a predefined small range. In implementations, an FSM can be selected based upon one or more criteria, including which FSM characteristics provide the best suppression for long transitions (e.g., fewer 3t and 4t transitions, and so forth). However, total number of transitions can also be used to select an FSM, and may not be easily observed from transition run length distributions. Thus, a smallest number of transitions may also be used to select an FSM. In a specific instance, for rates between at least approximately 0.9468 and 0.9614, code rates with simple integer ratios in the range of 17/18, 19/20, 20/21, 21/22, 22/23, 23/24, and 24/25 can be used. In this implementation, an FSM with mtr5 or longer run length can be obtained.
However, there is still a performance gap between the best performance of FSMs constructed as described above and the optimized Markov source. For example, by comparing the transition properties of an optimized Markov source and the FSMs constructed above, both the total number of transitions and the number of long transitions of the FSMs may be worse (e.g., more) than an optimized Markov source. Accordingly, ML-FSMs are described that can provide a more flexible structure than the ML-FSMs above. For example, multi-penalty can be implemented from time to time (e.g., in one period). In this manner, the ML-FSMs can be fine-tuned using a smaller granularity, and the transition distribution can be driven to an optimized distribution in many instances. In implementations, rather than use a uniform penalty on different time ticks, different sets of penalties can be assigned at different time ticks, on the transitions to be suppressed.
Referring to FIG. 3, an example ML run-length limited (RLL) FSM with two sets of penalties is illustrated with an accompanying timeline. In this example, the level of the FSM is equal to eight (8), the first penalty is equal to [1, 2, 3, 4], the second penalty is equal to [0, 1, 2, 3], and the drop is equal to five (5) (the drop is not shown in the accompanying figure, but may be included at, for example, t=period). In this figure, penLen1 is the time duration for application of the first penalty, penLen2 is the time duration for application of the second penalty, and the period is equal to penLen1 plus penLen2. In implementations, the ML-FSM can have one or more of the following characteristics. The capacity of the FSM can be greater than the code rate needed for a particular configuration. The transition distribution of the FSM can approach the optimized transition distribution. Further, the search complexity can be controlled or limited to a manageable range. For example, in some instances, only two sets of penalties are used (e.g., where penLen2 is equal to the period minus penLen1). In this configuration, the search is weeping penLen2 from one (1) to the period divided by two (2). The duration of each set of penalties is variable. In instances where the transition distribution computation is more time consuming than the capacity calculation, FSMs can be selected for a transition distribution test such that the selected FSMs have a capacity equal to or greater than the target rate and less than or equal to the target rate plus delta, where delta represents a small number (e.g., on the order of five one-hundredths (0.05)). Further, as the search is running with increased penalties in an inner loop, the loop can be broken when the capacity is less than the target rate.
As described herein, a search algorithm for implementing a multi-penalty implementation is provided using pseudo code, where [p1, p2, p3, p4] represents penalties for the first, second, third, and fourth transitions, respectively, which are the first set of penalties; [p21, p22, p23, p24] represents the second set of penalties; P_limit is a predefined limit for the values of the penalties; L_limit is the limit set for a particular level; and penLen2 is the duration for the second set of penalties. Then, for a particular period,












N_P1 = [p1, p2, p3, p4] * [1, P_limit, P_limit{circumflex over ( )}2, P_limit{circumflex over ( )}3]T;


N_P2 = [p21, p22, p23, p24] * [1, P_limit, P_limit{circumflex over ( )}2, P_limit{circumflex over ( )}3]T;


for (level = 2; level <= L_limit; level++) {


  for (drop = level − 1; drop > 0; drop−−) {


   for (pL2 = 1; pL2 <= penLen2; pL2++) {


     for (p = 1; p <= N_P1; p++) {


      for (p2 = 1; p2 <= N_P2; p2++) {


        construct the ML-FSM according to pL2, p, and p2


        compute the capacity of the FSM, Cfsm


        if Cfsm falls in the range [target rate, target


        rate + delta]then


         - compute the probabilities for dc, 1, 2, 3 and


           4 transition runs, i.e., the distribution


         - if the distribution is close to the optimized one,


           then this setting [level, drop, p1, p2] is saved


           to a list for later processing


        else


         - continue for the next pL2 value


      }


     }


   }


  }


}









After executing the preceding search algorithm, a list of possible FSMs is available. One or more FSMs can be selected according to, for instance, FSMs that provide sufficient large capacity, transition distributions that are close to an optimized transition, and so forth. For the purposes of the present disclosure, the term “close” can be used to refer to transitions with probabilities in a predefined small range. In implementations, an FSM can be selected based upon one or more criteria, including which FSM characteristics provide the best suppression for long transitions (e.g., fewer 3t and 4t transitions, and so forth). In some implementations, a smallest number of total transitions may be used to select an FSM. However, in other instances, a larger number of total transitions can be used to provide better suppression on long transitions (e.g., 3t and 4t) to achieve optimal performance. For example, in some instances 4t transitions may be even fewer than the optimized Markov source.
FIG. 4 illustrates a method 400 in an example implementation that may be employed by a read channel, such as the read channel circuit 510 of FIG. 5, to construct MTR modulation code. As shown, a magnetic recording channel is modeled as a partial response channel (Block 410). In some instances, the magnetic recording channel is modeled using AR modeling (Block 412). Then, a source of information to the magnetic recording channel is modeled to provide an optimized Markov source (Block 420). Next, MTR modulation code is constructed to mimic the optimized Markov source based upon an FSM having a limited transition run length and a multi-level periodic structure. In implementations, the FSM provides at least two different sets of penalties in a period (Block 430).
Although the techniques disclosed herein are not limited to any particular application, several examples of applications are presented in FIGS. 5 and 6. In FIG. 5 a storage system 500 is illustrated. The storage system 500 includes a read channel circuit 510 that employs information divergence based data processing circuitry in accordance with an example implementation of the present disclosure. The storage system 500 may be, for example, a hard disk drive (HDD). As shown, the storage system 500 includes a preamplifier 570, an interface controller 520, a hard disk controller 566, a motor controller 568, a spindle motor 572, a disk platter 578, and a read/write head assembly 576. The interface controller 520 controls addressing and timing of data to/from the disk platter 578, and interacts with a host controller that includes out of order constraint command circuitry. The data on the disk platter 578 includes groups of magnetic signals that may be detected by the read/write head assembly 576 when the assembly is properly positioned over disk platter 578. In one or more implementations, the disk platter 578 includes magnetic signals recorded in accordance with either a longitudinal or a perpendicular recording scheme.
In a typical read operation, the read/write head assembly 576 is accurately positioned by the motor controller 568 over a desired data track on the disk platter 578. The motor controller 568 positions the read/write head assembly 576 in relation to the disk platter 578 and drives the spindle motor 572 by moving the read/write head assembly 576 to the proper data track on the disk platter 578 under the direction of the hard disk controller 566. The spindle motor 572 spins the disk platter 578 at a determined spin rate (e.g., at a determined number of revolutions per minute (RPM)). Once the read/write head assembly 576 is positioned adjacent to the proper data track, magnetic signals representing data on the disk platter 578 are sensed by the read/write head assembly 576 as the disk platter 578 is rotated by the spindle motor 572. The sensed magnetic signals are provided as a continuous, minute analog signal representative of the magnetic data on the disk platter 578. This minute analog signal is transferred from the read/write head assembly 576 to the read channel circuit 510 via a preamplifier 570. The preamplifier 570 is operable to amplify the minute analog signals accessed from the disk platter 578. In turn, the read channel circuit 510 decodes and digitizes the received analog signal to recreate the information originally written to the disk platter 578. This data is provided as read data 503 to a receiving circuit. A write operation is substantially the opposite of the preceding read operation with write data 501 being provided to the read channel circuit 510. This data is then encoded and written to the disk platter 578.
It should be noted that the storage system 500 may be integrated into a larger storage system such as, for example, a RAID (redundant array of inexpensive disks or redundant array of independent disks) based storage system. RAID storage systems increase stability and reliability through redundancy, combining multiple disks as a logical unit. In this manner, data may be spread across a number of disks included in the RAID storage system according to a variety of algorithms and accessed by an operating system as if the RAID storage system were a single disk drive. For example, data may be mirrored to multiple disks in the RAID storage system, or may be sliced and distributed across multiple disks using a number of techniques. If a small number of disks in the RAID storage system fail or become unavailable, error correction techniques may be used to recreate the missing data based on the remaining portions of the data from the other disks in the RAID storage system. The disks in the RAID storage system may be, but are not necessarily limited to, individual storage systems such as storage system 500, and may be located in close proximity to each other or distributed more widely for increased security. In a write operation, write data is provided to a controller, which stores the write data across the disks, for example by mirroring or by striping the write data. In a read operation, the controller retrieves the data from the disks. The controller then yields the resulting read data as if the RAID storage system were a single disk drive.
A data decoder circuit used in relation to read channel circuit 510 may be, but is not necessarily limited to, a low density parity check (LDPC) decoder circuit. Low density parity check technology is applicable to transmission of information over various channels and/or information storage systems on various media. Transmission applications include, but are not necessarily limited to: optical fiber, radio frequency channels, wired or wireless local area networks, digital subscriber line technologies, wireless cellular, Ethernet over various mediums such as copper or optical fiber, cable channels such as cable television, and Earth-satellite communications. Storage applications include, but are not necessarily limited to: hard disk drives, compact disks, digital video disks, magnetic tapes and memory devices such as DRAM, NAND flash, NOR flash, other nonvolatile memories and solid state drives.
In addition, it should be noted that the storage system 500 may be configured to include solid state memory to store data in addition to the storage offered by the disk platter 578. Solid state memory may be used in parallel to the disk platter 578 to provide additional storage. In implementations, the solid state memory may receive and/or provide information directly to the read channel circuit 510. Additionally, the solid state memory may be used as a cache, e.g., to provide faster access time than that offered by the disk platter 578. In implementations, the solid state memory may be disposed between the interface controller 520 and the read channel circuit 510 and can operate as a pass through to the disk platter 578, e.g., when requested data is not available in the solid state memory and/or when the solid state memory does not have sufficient storage to hold a newly written data set. A variety of storage systems including disk platter 578 and solid state memory can be furnished in accordance with example implementations of the present disclosure.
Turning to FIG. 6, a data transmission system 600 including a receiver 620 having information divergence based data processing circuitry is shown in accordance with example implementations of the present disclosure. Data transmission system 600 includes a transmitter 610 that is operable to transmit encoded information via a transfer medium 630. The encoded data is received from transfer medium 630 by a receiver 620. Receiver 620 processes the received input to yield the originally transmitted data.
Generally, any of the functions described herein can be implemented using hardware (e.g., fixed logic circuitry such as integrated circuits), software, firmware, manual processing, or a combination of these implementations. Thus, the blocks discussed in the above disclosure generally represent hardware (e.g., fixed logic circuitry such as integrated circuits), software, firmware, or a combination thereof. In the instance of a hardware implementation, for instance, the various blocks discussed in the above disclosure may be implemented as integrated circuits along with other functionality. Such integrated circuits may include all of the functions of a given block, system or circuit, or a portion of the functions of the block, system or circuit. Further, elements of the blocks, systems or circuits may be implemented across multiple integrated circuits. Such integrated circuits may comprise various integrated circuits including, but not necessarily limited to: a monolithic integrated circuit, a flip chip integrated circuit, a multichip module integrated circuit, and/or a mixed signal integrated circuit. In the instance of a software implementation, for instance, the various blocks discussed in the above disclosure represent executable instructions (e.g., program code) that perform specified tasks when executed on a processor. These executable instructions can be stored in one or more tangible computer readable media. For example, the read channel 510 can employ a processor that receives information from the disk platter 578 and recovers data from the disk platter using MTR modulation code. The read channel 510 can also employ memory having computer executable instructions stored thereon, where the computer executable instructions are configured for execution by the processor to perform one or more of the techniques described herein. In some such instances, the entire system, block or circuit may be implemented using its software or firmware equivalent. In other instances, one part of a given system, block or circuit may be implemented in software or firmware, while other parts are implemented in hardware.
Although the subject matter has been described in language specific to structural features and/or methodological acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or acts described. Although various configurations are discussed the apparatus, systems, subsystems, components and so forth can be constructed in a variety of ways without departing from this disclosure. Rather, the specific features and acts are disclosed as example forms of implementing the claims.

## Claims

1. A system comprising:
a processor configured to receive information from a hard disk drive (HDD) via a read channel and recover data from the HDD using maximum transition run (MTR) modulation code; and
a memory having computer executable instructions stored thereon, the computer executable instructions configured for execution by the processor to:
model a magnetic recording channel as a partial response channel,
model a source of information to the magnetic recording channel to provide an optimized Markov source, and
construct an MTR modulation code to mimic the optimized Markov source based upon a finite state machine (FSM) having a limited transition run length and a multi-level periodic structure, the FSM providing at least two different sets of penalties in a period.

2. The system as recited in claim 1, wherein the magnetic recording channel is modeled using data dependent autoregressive (AR) modeling.

3. The system as recited in claim 1, wherein the multi-level periodic structure of the FSM has an output granularity of at least a block.

4. The system as recited in claim 1, wherein a constraint of the FSM is loosened during a final time increment of the period.

5. The system as recited in claim 4, wherein the constraint is loosened by lowering a group of connections by at least one level.

6. The system as recited in claim 1, wherein a capacity of the FSM is greater than a code rate for the magnetic recording channel.

7. The system as recited in claim 1, wherein a transition distribution of the FSM is at least substantially equal to a transition distribution of the optimized Markov source.

8. A non-transitory computer-readable storage medium having computer executable instructions configured to construct maximum transition run (MTR) modulation code, the computer executable instructions comprising:
modeling, by a processor, a source of information to a partial response channel to provide an optimized Markov source; and
constructing, by the processor, an MTR modulation code to mimic the optimized Markov source based upon a finite state machine (FSM) having a limited transition run length and a multi-level periodic structure, the FSM providing at least two different sets of penalties in a period.

9. The computer-readable storage medium as recited in claim 8, wherein the partial response channel models a magnetic recording channel using data dependent autoregressive (AR) modeling.

10. The computer-readable storage medium as recited in claim 8, wherein the multi-level periodic structure of the FSM has an output granularity of at least a block.

11. The computer-readable storage medium as recited in claim 8, wherein a constraint of the FSM is loosened during a final time increment of the period.

12. The computer-readable storage medium as recited in claim 11, wherein the constraint is loosened by lowering a group of connections by at least one level.

13. The computer-readable storage medium as recited in claim 8, wherein a transition distribution of the FSM is at least substantially equal to a transition distribution of the optimized Markov source.

14. A computer-implemented method for recovering data from a hard disk drive (HDD), the computer-implemented method comprising:
modeling, by a processor, a magnetic recording channel as a partial response channel;
modeling, by the processor, a source of information to the magnetic recording channel to provide an optimized Markov source; and
causing the processor to construct an MTR modulation code to mimic the optimized Markov source based upon a finite state machine (FSM) having a limited transition run length and a multi-level periodic structure, the FSM providing at least two different sets of penalties in a period.

15. The computer-implemented method as recited in claim 14, wherein the magnetic recording channel is modeled using data dependent autoregressive (AR) modeling.

16. The computer-implemented method as recited in claim 14, wherein the multi-level periodic structure of the FSM has an output granularity of at least a block.

17. The computer-implemented method as recited in claim 14, wherein a constraint of the FSM is loosened during a final time increment of the period.

18. The computer-implemented method as recited in claim 17, wherein the constraint is loosened by lowering a group of connections by at least one level.

19. The computer-implemented method as recited in claim 14, wherein a capacity of the FSM is greater than a code rate for the magnetic recording channel.

20. The computer-implemented method as recited in claim 14, wherein a transition distribution of the FSM is at least substantially equal to a transition distribution of the optimized Markov source.