Low-Density Arrays of Circulant Matrices: Rank and Row-Redundancy Analysis, and Quasi-Cyclic LDPC Codes

This paper is concerned with general analysis on the rank and row-redundancy of an array of circulants whose null space defines a QC-LDPC code. Based on the Fourier transform and the properties of conjugacy classes and Hadamard products of matrices, we derive tight upper bounds on rank and row-redundancy for general array of circulants, which make it possible to consider row-redundancy in constructions of QC-LDPC codes to achieve better performance. We further investigate the rank of two types of construction of QC-LDPC codes: constructions based on Vandermonde Matrices and Latin Squares and give combinatorial expression of the exact rank in some specific cases, which demonstrates the tightness of the bound we derive. Moreover, several types of new construction of QC-LDPC codes with large row-redundancy are presented and analyzed.

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

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

[3]  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.

[4]  P. B. Bhattacharya,et al.  First Course In Linear Algebra , 1983 .

[5]  Yuan Zhou.  Introduction to Coding Theory , 2010 .

[6]  Shu Lin,et al.  Codes on finite geometries , 2005, IEEE Transactions on Information Theory.

[7]  J.L. Massey,et al.  Theory and practice of error control codes , 1986, Proceedings of the IEEE.

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

[9]  Shu Lin,et al.  A class of low-density parity-check codes constructed based on Reed-Solomon codes with two information symbols , 2003, IEEE Communications Letters.

[10]  Shu Lin,et al.  Error control coding : fundamentals and applications , 1983 .

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

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

[13]  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.

[14]  Robert Michael Tanner,et al.  A transform theory for a class of group-invariant codes , 1988, IEEE Trans. Inf. Theory.

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

[16]  Qiuju Diao,et al.  Cyclic and Quasi-Cyclic LDPC Codes on Constrained Parity-Check Matrices and Their Trapping Sets , 2012, IEEE Transactions on Information Theory.

[17]  Tadao Kasami.  A Gilbert-Varshamov bound for quasi-cycle codes of rate 1/2 (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[18]  R. Blahut.  Theory and practice of error control codes , 1983 .

[19]  R. Townsend,et al.  Self-orthogonal quasi-cyclic codes , 1967, IEEE Trans. Inf. Theory.

[20]  Khaled A. S. Abdel-Ghaffar,et al.  A unified approach to the construction of binary and nonbinary quasi-cyclic LDPC codes based on finite fields , 2009, IEEE Transactions on Communications.

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

[22]  Norifumi Kamiya.  High-rate quasi-cyclic low-density parity-check codes derived from finite affine planes , 2005, ISIT.

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

[24]  Shu Lin,et al.  Construction of Regular and Irregular LDPC Codes: Geometry Decomposition and Masking , 2007, IEEE Transactions on Information Theory.

[25]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[26]  Rüdiger L. Urbanke,et al.  Design of capacity-approaching irregular low-density parity-check codes , 2001, IEEE Trans. Inf. Theory.

[27]  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.

[28]  Qiuju Diao,et al.  A transform approach for computing the ranks of parity-check matrices of quasi-cyclic LDPC codes , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[29]  David J. C. MacKay,et al.  Good Error-Correcting Codes Based on Very Sparse Matrices , 1997, IEEE Trans. Inf. Theory.

[30]  M. Darnell,et al.  Error Control Coding: Fundamentals and Applications , 1985 .

[31]  Zhongfeng Wang,et al.  Low-Complexity High-Speed Decoder Design for Quasi-Cyclic LDPC Codes , 2007, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[32]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .

[33]  Zhi Ding,et al.  Quasi-Cyclic LDPC Codes on Cyclic Subgroups of Finite Fields , 2011, IEEE Transactions on Communications.

[34]  WangZhongfeng,et al.  Low-complexity high-speed decoder design for quasi-cyclic LDPC codes , 2007 .

[35]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[36]  Norifumi Kamiya,et al.  High-Rate Quasi-Cyclic Low-Density Parity-Check Codes Derived From Finite Affine Planes , 2005, IEEE Transactions on Information Theory.