Finite-Length Algebraic Spatially-Coupled Quasi-Cyclic LDPC Codes

The replicate-and-mask (R&M) construction of finite-length spatially-coupled (SC) LDPC codes is proposed in this paper. The proposed R&M construction generalizes the conventional matrix unwrapping construction and contains it as a special case. The R&M construction of a class of algebraic spatially coupled (SC) quasi-cyclic (QC) LDPC codes over arbitrary finite fields is demonstrated. The girth, rank, and time-varying periodicity of the proposed R&M SC QC LDPC codes are analyzed. The error rate performance of finite-length nonbinary algebraic SC QC LDPC codes is investigated with window decoding. Compared to the conventional unwrapping construction, it is found through numerical simulations that the R&M construction resulted in SC QC LDPC codes with better block error rate performance and lower error floors. With a flooding schedule decoder, it is shown that the proposed R&M algebraic SC QC LDPC codes have better error performance than the corresponding LDPC block codes and random SC codes. The R&M construction of irregular SC QC LDPC codes is demonstrated. It is shown that low-complexity regular puncturing schemes can be deployed on these codes to construct families of rate-compatible irregular SC QC LDPC codes with good performance.

[1]  Khaled A. S. Abdel-Ghaffar,et al.  A Matrix-Theoretic Approach to the Construction of Non-Binary Quasi-Cyclic LDPC Codes , 2015, IEEE Transactions on Communications.

[2]  Paul H. Siegel,et al.  Windowed Decoding of Protograph-Based LDPC Convolutional Codes Over Erasure Channels , 2010, IEEE Transactions on Information Theory.

[3]  Michael Lentmaier,et al.  To the Theory of Low-Density Convolutional Codes. II , 2001, Probl. Inf. Transm..

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

[5]  Keshab K. Parhi,et al.  Overlapped message passing for quasi-cyclic low-density parity check codes , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[6]  Tingting Liang,et al.  Efficient Encoding of Quasi-Cyclic Low-Density Parity-Check Codes , 2018, 2018 IEEE 3rd Advanced Information Technology, Electronic and Automation Control Conference (IAEAC).

[7]  Lara Dolecek,et al.  Non-Binary Protograph-Based LDPC Codes: Enumerators, Analysis, and Designs , 2014, IEEE Transactions on Information Theory.

[8]  Xiao Ma,et al.  Performance Comparison of LDPC Block and Spatially Coupled Codes Over GF $(q)$ , 2014, IEEE Transactions on Communications.

[9]  Qiuju Diao,et al.  A matrix-theoretic approach for analyzing quasi-cyclic low-density parity-check codes , 2012, IEEE Transactions on Information Theory.

[10]  William E. Ryan,et al.  An unnoticed strong connection between algebraic-based and protograph-based LDPC codes, Part I: Binary case and interpretation , 2015, 2015 Information Theory and Applications Workshop (ITA).

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

[12]  Rüdiger L. Urbanke,et al.  Threshold saturation on BMS channels via spatial coupling , 2010, 2010 6th International Symposium on Turbo Codes & Iterative Information Processing.

[13]  Rudiger Urbanke,et al.  Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the BEC , 2010, ISIT.

[14]  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.

[15]  Rüdiger L. Urbanke,et al.  Spatially coupled ensembles universally achieve capacity under belief propagation , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[16]  Shu Lin,et al.  Low-density parity-check codes based on finite geometries: A rediscovery and new results , 2001, IEEE Trans. Inf. Theory.

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

[18]  Giulio Colavolpe,et al.  Nonbinary spatially-coupled LDPC codes on the binary erasure channel , 2013, 2013 IEEE International Conference on Communications (ICC).

[19]  Ali Emre Pusane,et al.  Deriving Good LDPC Convolutional Codes from LDPC Block Codes , 2010, IEEE Transactions on Information Theory.

[20]  Henry D. Pfister,et al.  A simple proof of threshold saturation for coupled scalar recursions , 2012, 2012 7th International Symposium on Turbo Codes and Iterative Information Processing (ISTC).

[21]  Kenta Kasai,et al.  Design and Performance of Rate-Compatible Non-binary LDPC Convolutional Codes , 2010, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[22]  Shu Lin,et al.  Construction of Quasi-Cyclic LDPC Codes for AWGN and Binary Erasure Channels: A Finite Field Approach , 2007, IEEE Transactions on Information Theory.

[23]  Khaled A. S. Abdel-Ghaffar,et al.  Algebraic Quasi-Cyclic LDPC Codes: Construction, Low Error-Floor, Large Girth and a Reduced-Complexity Decoding Scheme , 2014, IEEE Transactions on Communications.

[24]  Jungwon Lee,et al.  Non-binary algebraic spatially-coupled quasi-cyclic LDPC codes , 2014, 2014 IEEE International Symposium on Information Theory.

[25]  Sarah J. Johnson,et al.  Memory Efficient Decoders using Spatially Coupled Quasi-Cyclic LDPC Codes , 2013, ArXiv.

[26]  David G. M. Mitchell,et al.  Quasi-cyclic LDPC codes based on pre-lifted protographs , 2011, ITW.

[27]  William Ryan,et al.  Channel Codes by William Ryan , 2009 .

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

[29]  Shu Lin,et al.  Channel Codes: Classical and Modern , 2009 .

[30]  Michael Lentmaier,et al.  Iterative Decoding Threshold Analysis for LDPC Convolutional Codes , 2010, IEEE Transactions on Information Theory.

[31]  Lijun Zhang,et al.  Hybrid construction of LDPC codes with (14, 8) Hamming code , 2015, 2015 IEEE 9th International Conference on Anti-counterfeiting, Security, and Identification (ASID).

[32]  Jilei Hou,et al.  Design of rate-compatible structured LDPC codes for hybrid ARQ applications , 2009, IEEE Journal on Selected Areas in Communications.

[33]  Zongwang Li,et al.  Efficient encoding of quasi-cyclic low-density parity-check codes , 2006, IEEE Trans. Commun..

[34]  Dariush Divsalar,et al.  Constructing LDPC codes from simple loop-free encoding modules , 2005, IEEE International Conference on Communications, 2005. ICC 2005. 2005.

[35]  Hideki Imai,et al.  Spatially coupled quasi-cyclic quantum LDPC codes , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[36]  Rüdiger L. Urbanke,et al.  Efficient encoding of low-density parity-check codes , 2001, IEEE Trans. Inf. Theory.

[37]  Toshiaki Koike-Akino,et al.  Threshold analysis of non-binary spatially-coupled LDPC codes with windowed decoding , 2014, 2014 IEEE International Symposium on Information Theory.

[38]  Qin Huang,et al.  Low-Density Arrays of Circulant Matrices: Rank and Row-Redundancy Analysis, and Quasi-Cyclic LDPC Codes , 2012, ArXiv.

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

[40]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[41]  Michael Lentmaier,et al.  Quasi-cyclic asymptotically regular LDPC codes , 2010, 2010 IEEE Information Theory Workshop.

[42]  Marc P. C. Fossorier,et al.  Quasi-Cyclic Low-Density Parity-Check Codes From Circulant Permutation Matrices , 2004, IEEE Trans. Inf. Theory.

[43]  David G. M. Mitchell,et al.  Robust Rate-Compatible Punctured LDPC Convolutional Codes , 2013, IEEE Transactions on Communications.

[44]  Brendan J. Frey,et al.  Iterative Decoding of Compound Codes by Probability Propagation in Graphical Models , 1998, IEEE J. Sel. Areas Commun..

[45]  William Ryan,et al.  Channel Codes: Classical and Modern , 2009 .