Theory and Practice of Non-Binary Graph-Based Codes: A Combinatorial View

We are undergoing a revolution in data. The ever-growing amount of information in our world has created an unprecedented demand for ultra-reliable, affordable, and resource-efficient data storage systems. Error-correcting codes, as a critical component of any memory device, will play a crucial role in the future of data storage.One particular class of error-correcting codes, known as graph-based codes, has drawn significant attention in both academia and in industry. Graph-based codes offer superior performance compared to traditional algebraic codes. Recently, it has been shown that non-binary graph-based codes, which operate over finite fields rather than binary alphabets, outperform their binary counterparts and exhibit outstanding overall performance. For this reason, these codes are particularly suitable for emerging data storage systems.In this dissertation, we present a comprehensive combinatorial analysis of non-binary graph-based codes. We perform both finite-length and asymptotic analyses for these codes, providing a systematic framework to evaluate and optimize various families of non-binary graph-based codes.In the finite-length case, we provide a mathematical characterization of the error floor problem, including a general definition of absorbing sets over non-binary alphabets. We consider several structured low-density parity-check (LDPC) codes, including quasi-cyclic and spatially-coupled codes, as well as unstructured LDPC codes. We offer design guidelines for non-binary LDPC codes with outstanding performance in extremely low error-rate regimes; making them excellent candidates for data storage applications.In the asymptotic case, we provide a novel toolbox for the evaluation of families of non-binary graph-based codes. By utilizing insights from graph theory and combinatorics, we establish enumerators for a general family of graph-based codes which are constructed based on protographs. We provide asymptotic distributions of codewords and trapping sets for the family of protograph-based codes. Furthermore, we present an asymptotic enumeration of binary and non-binary elementary absorbing sets for regular code ensembles.The contributions of this dissertation can potentially impact a broad range of data storage and communication technologies that require excellent performance in high-reliability regimes.

[1]  Lara Dolecek,et al.  On the typical minimum distance of protograph-based non-binary LDPC codes , 2012, 2012 Information Theory and Applications Workshop.

[2]  Jörg Kliewer,et al.  Trapping set enumerators for repeat multiple accumulate code ensembles , 2009, 2009 IEEE International Symposium on Information Theory.

[3]  Wuyang Zhou,et al.  Large-Girth Nonbinary QC-LDPC Codes of Various Lengths , 2010, IEEE Transactions on Communications.

[4]  Kenta Kasai,et al.  Analysis of stopping constellation distribution for irregular non-binary LDPC code ensemble , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[5]  Shu Lin,et al.  Transactions Papers - Constructions of Nonbinary Quasi-Cyclic LDPC Codes: A Finite Field Approach , 2008, IEEE Transactions on Communications.

[6]  Paul H. Siegel,et al.  On the capacity of finite state channels and the analysis of convolutional accumulate-m codes , 2003 .

[7]  D. Mackay,et al.  Low density parity check codes over GF(q) , 1998, 1998 Information Theory Workshop (Cat. No.98EX131).

[8]  Rüdiger L. Urbanke,et al.  Modern Coding Theory , 2008 .

[9]  Qin Huang,et al.  Quasi-Cyclic LDPC Codes: An Algebraic Construction, Rank Analysis, and Codes on Latin Squares , 2010, IEEE Transactions on Communications.

[10]  Lara Dolecek,et al.  Non-binary protograph-based LDPC codes for short block-lengths , 2012, 2012 IEEE Information Theory Workshop.

[11]  Khaled A. S. Abdel-Ghaffar,et al.  Algebraic construction of quasi-cyclic LDPC codes for the AWGN and erasure channels , 2006, IEEE Transactions on Communications.

[12]  Kamil Sh. Zigangirov,et al.  Time-varying periodic convolutional codes with low-density parity-check matrix , 1999, IEEE Trans. Inf. Theory.

[13]  Judea Pearl,et al.  Reverend Bayes on Inference Engines: A Distributed Hierarchical Approach , 1982, AAAI.

[14]  Radford M. Neal,et al.  Near Shannon limit performance of low density parity check codes , 1996 .

[15]  David Declercq,et al.  Decoding Algorithms for Nonbinary LDPC Codes Over GF$(q)$ , 2007, IEEE Transactions on Communications.

[16]  Robert Michael Tanner,et al.  A recursive approach to low complexity codes , 1981, IEEE Trans. Inf. Theory.

[17]  Simon Litsyn,et al.  On ensembles of low-density parity-check codes: Asymptotic distance distributions , 2002, IEEE Trans. Inf. Theory.

[18]  Alexandre Graell i Amat,et al.  Threshold Saturation for Nonbinary SC-LDPC Codes on the Binary Erasure Channel , 2016, IEEE Transactions on Information Theory.

[19]  H. Vincent Poor,et al.  Finding All Small Error-Prone Substructures in LDPC Codes , 2009, IEEE Transactions on Information Theory.

[20]  Lara Dolecek,et al.  Asymptotic distribution of absorbing sets and fully absorbing sets for regular sparse code ensembles , 2011 .

[21]  Lara Dolecek,et al.  Enumerators for protograph-based ensembles of nonbinary LDPC codes , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[22]  Lara Dolecek,et al.  On absorbing sets of structured sparse graph codes , 2010, 2010 Information Theory and Applications Workshop (ITA).

[23]  Lara Dolecek,et al.  Analysis of Absorbing Sets and Fully Absorbing Sets of Array-Based LDPC Codes , 2009, IEEE Transactions on Information Theory.

[24]  F. Moore,et al.  Polynomial Codes Over Certain Finite Fields , 2017 .

[25]  Stefano Valle,et al.  NB-LDPC: Absorbing set and importance sampling , 2012, 2012 7th International Symposium on Turbo Codes and Iterative Information Processing (ISTC).

[26]  William E. Ryan,et al.  Enumerators for Protograph-Based Ensembles of LDPC and Generalized LDPC Codes , 2011, IEEE Transactions on Information Theory.

[27]  Richard W. Hamming,et al.  Error detecting and error correcting codes , 1950 .

[28]  Los Angeles,et al.  Non-Binary Protograph-Based LDPC Codes: Analysis, Enumerators and Designs , 2013 .

[29]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[30]  Dariush Divsalar,et al.  Serial Concatenation of Interleaved Codes: Performance Analysis, Design, and Iterative Decoding , 1997, IEEE Trans. Inf. Theory.

[31]  Dwijendra K. Ray-Chaudhuri,et al.  Binary mixture flow with free energy lattice Boltzmann methods , 2022, arXiv.org.

[32]  Kenta Kasai,et al.  Error floors of non-binary LDPC codes , 2010, 2010 IEEE International Symposium on Information Theory.

[33]  Mostafa El-Khamy,et al.  New Approaches to the Analysis and Design of Reed-Solomon Related Codes , 2007 .

[34]  Amir H. Banihashemi,et al.  An efficient algorithm for finding dominant trapping sets of LDPC codes , 2011, 2010 6th International Symposium on Turbo Codes & Iterative Information Processing.

[35]  Emina Soljanin,et al.  Asymptotic Spectra of Trapping Sets in Regular and Irregular LDPC Code Ensembles , 2007, IEEE Transactions on Information Theory.

[36]  Simon Litsyn,et al.  Design of non-binary quasi-cyclic LDPC codes by ACE optimization , 2013, 2013 IEEE Information Theory Workshop (ITW).

[37]  Kenta Kasai,et al.  Analysis of error floors of generalized non-binary LDPC codes over q-ary memoryless symmetric channels , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[38]  Naresh R. Shanbhag,et al.  High-throughput LDPC decoders , 2003, IEEE Trans. Very Large Scale Integr. Syst..

[39]  Edward A. Bender,et al.  The asymptotic number of non-negative integer matrices with given row and column sums , 1974, Discret. Math..

[40]  David Declercq,et al.  Low-complexity decoding for non-binary LDPC codes in high order fields , 2010, IEEE Transactions on Communications.

[41]  Evangelos Eleftheriou,et al.  Regular and irregular progressive edge-growth tanner graphs , 2005, IEEE Transactions on Information Theory.

[42]  Rong-Rong Chen,et al.  Design of Nonbinary Quasi-Cyclic LDPC Cycle Codes , 2007, 2007 IEEE Information Theory Workshop.

[43]  Iryna Andriyanova,et al.  Quantum LDPC codes obtained by non-binary constructions , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[44]  Shuai Zhang,et al.  Causes and dynamics of LDPC error floors on AWGN channels , 2011, 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[45]  J. Thorpe.  Low-Density Parity-Check (LDPC) Codes Constructed from Protographs , 2003 .

[46]  Kenta Kasai,et al.  Analysis of Error Floors of Non-Binary LDPC Codes over MBIOS Channel , 2011, ICC.

[47]  Jörg Kliewer,et al.  Coding Theorems for Repeat Multiple Accumulate Codes , 2008, ArXiv.

[48]  A. Glavieux,et al.  Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1 , 1993, Proceedings of ICC '93 - IEEE International Conference on Communications.

[49]  Shengli Zhou,et al.  Nonbinary LDPC Coding for Multicarrier Underwater Acoustic Communication , 2008, IEEE Journal on Selected Areas in Communications.

[50]  Iryna Andriyanova,et al.  Some Results on MAP Decoding of Non-Binary LDPC Codes Over the BEC , 2011, IEEE Transactions on Information Theory.

[51]  Jingyu Kang,et al.  An Iterative Decoding Algorithm with Backtracking to Lower the Error-Floors of LDPC Codes , 2011, IEEE Transactions on Communications.

[52]  Lara Dolecek,et al.  Absorbing set characterization of array-based spatially coupled LDPC codes , 2014, 2014 IEEE International Symposium on Information Theory.

[53]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[54]  Qin Huang,et al.  Quasi-cyclic LDPC codes: an algebraic construction , 2010, IEEE Transactions on Communications.

[55]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .