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21
B.H.: Tradeoff functions for constrained systems with unconstrained positions
 IEEE Trans. Inf. Theory
, 2006
"... Abstract—We introduce a new method for analyzing and constructing combined modulation and errorcorrecting codes (ECCs), in particular codes that utilize some form of reverse concatenation and whose ECC decoding scheme requires easy access to soft information. We expand the work of Immink and Wijnga ..."
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Abstract—We introduce a new method for analyzing and constructing combined modulation and errorcorrecting codes (ECCs), in particular codes that utilize some form of reverse concatenation and whose ECC decoding scheme requires easy access to soft information. We expand the work of Immink and Wijngaarden and also of Campello, Marcus, New, and Wilson, in which certain bit positions in the modulation code are deliberately left unconstrained for the ECC parity bits, in the sense that such positions can take on either bit value without violating the constraint. Our method of analysis involves creating a single graph that incorporates information on these unconstrained positions directly into the constraint graph without any assumptions of periodicity or sets of unconstrained positions, and is thus completely general. We establish several properties of the tradeoff function that relates the density of unconstrained positions to the maximum code rate. In particular, the tradeoff function is shown to be concave and continuous. Algorithms for computing lower and upper bounds for this function are presented. We also show how to compute the maximum possible density of unconstrained positions and give explicit values for the runlengthlimited (RLL ()) and maximumtransitionrun (MTR ()) constraints. Index Terms—Bitinsertion schemes, bitstuffing, combined modulation and errorcorrecting codes (ECCs), maximum transition run (MTR) system, runlengthlimited (RLL) system, tradeoff functions, unconstrained positions. I.
Constraint gain
 IEEE Trans. Inform. Theory
, 2001
"... Abstract—In digital storage systems where the input to the noisy channel is required to satisfy a modulation constraint, the constrained code and errorcontrol code (ECC) are typically designed and decoded independently. The achievable rate for this situation is evaluated as the rate of average inte ..."
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Cited by 7 (1 self)
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Abstract—In digital storage systems where the input to the noisy channel is required to satisfy a modulation constraint, the constrained code and errorcontrol code (ECC) are typically designed and decoded independently. The achievable rate for this situation is evaluated as the rate of average intersection of the constraint and the ECC. The gap from the capacity of the noisy constrained channel is called the constraint gain, which represents the potential improvement in combining the design and decoding of the constrained code and the ECC. The constraint gain is computed for various constraints over the binaryinput additive white Gaussian noise (AWGN) channel (BIAWGNC) as well as over intersymbol interference (ISI) channels. Finally, it is shown that an infinite cascade of reverse concatenation with independent decoding of constraint and ECC has a capacity equal to the rate of average intersection. Index Terms—Capacity, constrained codes, errorcontrol codes (ECCs), noisy constrained channels, reverse concatenation. I.
Presentations of constrained systems with unconstrained positions
 IEEE Trans. Inform. Theory
"... Abstract — We give a polynomialtime construction of the set of sequences that satisfy a finitememory constraint defined by a finite list of forbidden blocks, with a specified set of bit positions unconstrained. Such a construction can be used to build modulation/errorcorrection codes (ECC codes) ..."
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Abstract — We give a polynomialtime construction of the set of sequences that satisfy a finitememory constraint defined by a finite list of forbidden blocks, with a specified set of bit positions unconstrained. Such a construction can be used to build modulation/errorcorrection codes (ECC codes) like the ones defined by the ImminkWijngaarden scheme in which certain bit positions are reserved for ECC parity. We give a lineartime construction of a finitestate presentation of a constrained system defined by a periodic list of forbidden blocks. These systems, called periodicfinitetype systems, were introduced by Moision and Siegel. Finally, we present a lineartime algorithm for constructing the minimal periodic forbidden blocks of a finite sequence for a given period. Index Terms — Directed acyclic word graph (DAWG), finitememory systems, finitestate encoders, forbidden blocks, maximum transition run (MTR) codes, modulation codes, periodicfinitetype (PFT) systems, runlength limited (RLL) codes. I.
Timevarying maximum transition run constraints
 IEEE Trans. Inform. Theory
, 2006
"... Abstract—Maximum transition run (MTR) constrained systems are used to improve detection performance in storage channels. Recently, there has been a growing interest in timevarying MTR (TMTR) systems, after such codes were observed to eliminate certain error events and thus provide high coding gain ..."
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Abstract—Maximum transition run (MTR) constrained systems are used to improve detection performance in storage channels. Recently, there has been a growing interest in timevarying MTR (TMTR) systems, after such codes were observed to eliminate certain error events and thus provide high coding gain for E n PR4 channels for n =2; 3. In this work, TMTR constraints parameterized by a vector, whose coordinates specify periodically the maximum runlengths of 1’s ending at the positions, are investigated. A canonical way to classify such constraints and simplify their minimal graph presentations is introduced. It is shown that there is a particularly simple presentation for a special class of TMTR constraints and explicit descriptions of their characteristic equations are derived. New upper bounds on the capacity of TMTR constraints are established, and an explicit linear ordering by capacity of all tight TMTR constraints up to period 4 is given. For MTR constrained systems with unconstrained positions, it is shown that the set of sequences restricted to the constrained positions yields a natural TMTR constraint. Using TMTR constraints, a new upper bound on the tradeoff function for MTR systems that relates the density of unconstrained positions to the maximum code rates is determined. Index Terms—Capacity, constrained systems, maximum transition run (MTR), time varying, tradeoff function, upper bounds. I.
2004. “Error Correction Coding for a Serial Digital MultiGigabit Communication System
 the proceedings for the IEEE ElectroInformation Technology (EIT) conference
"... An error correction code (ECC) design for a wired serial digital multigigabit communication system is presented. The code design combines a maximum run length code and a 2errorcorrecting primitive BCH code. The implementation of the design in a field programmable gate array (FPGA) and the logic d ..."
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An error correction code (ECC) design for a wired serial digital multigigabit communication system is presented. The code design combines a maximum run length code and a 2errorcorrecting primitive BCH code. The implementation of the design in a field programmable gate array (FPGA) and the logic design of this code for low latency is discussed. Results are presented showing the effectiveness of the code design for some backplane trace lengths and levels of equalization, and ineffectiveness for others. The work presented serves as the foundation for future investigations in the use of ECC in these systems.
Maximum insertion rate and capacity of multidimensional constraints
 IEEE International Symposium on Information Theory
"... Abstract — The maximum insertion rate of a onedimensional constrained system over a finite alphabet is defined to be the maximum density of positions that can be freely, and independently, filled in with arbitrary symbols of the alphabet and still satisfy the constraint. In this paper, this concept ..."
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Abstract — The maximum insertion rate of a onedimensional constrained system over a finite alphabet is defined to be the maximum density of positions that can be freely, and independently, filled in with arbitrary symbols of the alphabet and still satisfy the constraint. In this paper, this concept is extended to higher dimensional constraints, that is, to constraints on Ddimensional arrays defined by imposing a 1dimensional constraint in each dimension. We give a simple upper bound on the Ddimensional maximum insertion rate in terms of the individual 1dimensional maximum insertion rate. For Ddimensional constraints defined by imposing the same 1dimensional constraint in each dimension, we show that the Ddimensional maximum insertion rate is the same as the 1dimensional maximum insertion rate. In this case (called the isotropic or, sometimes, symmetric case), we show that the maximum insertion rate is a lower bound on the limiting Ddimensional capacity as D tends to infinity. Finally, we show that in the case of a finite memory constraint, when the maximum insertion rate is zero, the Ddimensional capacity decays exponentially fast to zero. I.
Simple classes of constrained systems with unconstrained positions that outperform the maxentropic bound
, 2008
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Tensorproduct parity codes: combination with constrained codes and application to perpendicular recording
 IEEE Trans. on Magnetics
, 2006
"... A parity code and a distance enhancing constrained code are often concatenated with a Reed–Solomon code to form a coding system for magnetic recording. The tensorproduct parity coding scheme helps to improve efficiency of the parity code while retaining the same level of performance. In this paper, ..."
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A parity code and a distance enhancing constrained code are often concatenated with a Reed–Solomon code to form a coding system for magnetic recording. The tensorproduct parity coding scheme helps to improve efficiency of the parity code while retaining the same level of performance. In this paper, we present two methods for combining a tensorproduct parity code with a distanceenhancing constrained code. The first method incorporates a constrained code with unconstrained positions. The second method uses a new technique, which we call wordset partitioning, to achieve a higher code rate relative to the first method. We simulate the performance of several coding systems based upon the two combination methods on a perpendicular recording channel, and we compare their symbol error rates and sector error rates with those of a system that uses only a Reed–Solomon code. Index Terms—Constrained coding, parity code, perpendicular magnetic recording, tensorproduct code. I.
Scaling taperecording areal densities to 100 Gb/in 2
 IBM JOURNAL OF RESEARCH AND DEVELOPMENT
, 2008
"... We examine the issue of scaling magnetic taperecording to higher areal densities, focusing on the challenges of achieving 100 Gb/in 2 in the linear tape format. The current highest achieved areal density demonstrations of 6.7 Gb/in 2 in the linear tape and 23.0 Gb/in 2 in the helical scan format pr ..."
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Cited by 1 (1 self)
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We examine the issue of scaling magnetic taperecording to higher areal densities, focusing on the challenges of achieving 100 Gb/in 2 in the linear tape format. The current highest achieved areal density demonstrations of 6.7 Gb/in 2 in the linear tape and 23.0 Gb/in 2 in the helical scan format provide a reference for this assessment. We argue that controlling the head–tape interaction is key to achieving high linear density, whereas trackfollowing and reeltoreel servomechanisms as well as transverse dimensional stability are key for achieving high track density. We envision that advancements in media, datadetection techniques, reeltoreel control, and lateral motion control will enable much higher areal densities. An achievable goal is a linear density of 800 Kb/in and a track pitch of 0.2 lm, resulting in an areal density of 100 Gb/in 2. A. J. Argumedo
The Art of Combining DistanceEnhancing Constrained Codes with ParityCheck Codes for Data Storage Channels 1
"... A general and systematic code design methodology is proposed to efficiently combine constrained codes with PC codes for data storage channels. The proposed constrained PC code includes two component codes: the normal constrained (NC) code and the parityrelated constrained (PRC) code. The NC code ca ..."
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A general and systematic code design methodology is proposed to efficiently combine constrained codes with PC codes for data storage channels. The proposed constrained PC code includes two component codes: the normal constrained (NC) code and the parityrelated constrained (PRC) code. The NC code can be any distanceenhancing constrained code, such as the maximum transition run (MTR) code or repeated minimum transition runlength (RMTR) code. The PRC code can be any linear binary PC code. The constrained PC codes can be designed either in nonreturntozeroinverse (NRZI) format or nonreturntozero (NRZ) format. The rates of the designed codes are only a few tenths of a percent below the theoretical maximum. The proposed code design method enables soft information to be available to the PC decoder and facilitates soft decoding of PC codes. Furthermore, since errors are corrected equally well over the entire constrained PC codeword, error propagation due to parity bits is avoided. Efficient finitestate encoding methods are proposed to design capacityapproaching constrained codes and constrained PC codes with RMTR or MTR constraint. The generality and efficiency of the proposed code design methodology are shown by various code design examples for both magnetic and optical recording channels. Index Terms Distanceenhancing constrained codes, error correction codes (ECCs), paritycheck codes, finitestate encoding method, postprocessor, soft decoding. 2 I.