A Novel Coding Scheme for Encoding and Iterative Soft-Decision Decoding of Binary BCH Codes of Prime Lengths

A novel coding scheme is presented for encoding and iterative soft-decision decoding of binary BCH codes of prime lengths. The encoding of such a BCH code is performed on a collection of codewords which are mapped through Galois Fourier transform into a codeword of a nonbinary low-density parity-check (LDPC) code which has a binary parity-check matrix for transmission. Using this matrix, a binary iterative soft-decision decoding algorithm based on belief-propagation is applied to jointly decode a collection of codewords of the BCH code. The joint-decoding allows information sharing among the received vectors corresponding to the codewords in the collection during the iterative decoding process. For decoding a BCH code of prime length, the proposed decoding scheme not only requires low decoding complexity, but also yields superior performance. The proposed decoding scheme can achieve a joint-decoding gain over the maximum likelihood decoding of individual codewords.

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

[2]  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.

[3]  Qiuju Diao,et al.  LDPC Codes on Partial Geometries: Construction, Trapping Set Structure, and Puncturing , 2013, IEEE Transactions on Information Theory.

[4]  Hideki Imai,et al.  Reduced complexity iterative decoding of low-density parity check codes based on belief propagation , 1999, IEEE Trans. Commun..

[5]  Jinghu Chen,et al.  Near optimum universal belief propagation based decoding of low-density parity check codes , 2002, IEEE Trans. Commun..

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

[7]  Shu Lin,et al.  Iterative soft-decision decoding of reed-solomon codes of prime lengths , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[8]  Qiuju Diao,et al.  A transform approach for analyzing and constructing quasi-cyclic low-density parity-check codes , 2011, 2011 Information Theory and Applications Workshop.

[9]  Sen M. Kuo,et al.  Fast Fourier Transform and Its Applications , 2002 .

[10]  D.J.C. MacKay,et al.  Good error-correcting codes based on very sparse matrices , 1997, Proceedings of IEEE International Symposium on Information Theory.

[11]  W. W. Peterson,et al.  Encoding and error-correction procedures for the Bose-Chaudhuri codes , 1960, IRE Trans. Inf. Theory.

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

[13]  David Declercq,et al.  Fast decoding algorithm for LDPC over GF(2/sup q/) , 2003, Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674).

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

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

[16]  James L. Massey,et al.  Shift-register synthesis and BCH decoding , 1969, IEEE Trans. Inf. Theory.

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

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

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

[20]  Ron M. Roth,et al.  Introduction to Coding Theory , 2019, Discrete Mathematics.