On the Stopping Distance of Array Code Parity-Check Matrices

For q an odd prime and 1 les m les q, we study two binary qm times q2 parity check matrices for binary array codes. For both parity check matrices, we determine the stopping distance and the minimum distance of the associated code for 2 les m les 3, and for (m, q)=(4, 5). In the case (m, q)=(4, 7), the stopping distance and the related minimum distance are also determined for one of the given parity check matrices. Moreover, we give a lower bound on the stopping distances for m > 3 and q > 3.

[1]  A. Vardy,et al.  Stopping sets in codes from designs , 2003, IEEE International Symposium on Information Theory, 2003. Proceedings..

[2]  Morteza Esmaeili,et al.  On the Stopping Distance and Stopping Redundancy of Product Codes , 2008, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[3]  Navin Kashyap,et al.  Shortened Array Codes of Large Girth , 2005, IEEE Transactions on Information Theory.

[4]  Dale E. Hocevar.  Efficient encoding for a family of quasi-cyclic LDPC codes , 2003, GLOBECOM '03. IEEE Global Telecommunications Conference (IEEE Cat. No.03CH37489).

[5]  Bane V. Vasic,et al.  High-rate girth-eight low-density parity-check codes on rectangular integer lattices , 2004, IEEE Transactions on Communications.

[6]  Eisaku Sasaki,et al.  Efficient encoding of QC-LDPC codes related to cyclic MDS codes , 2008, IEEE Journal on Selected Areas in Communications.

[7]  Shu-Tao Xia,et al.  On the stopping distance of finite geometry LDPC codes , 2006, IEEE Communications Letters.

[8]  X.-L. Li,et al.  A Family of Generalized Constant Modulus Algorithms for Blind Equalization , 2006, IEEE Transactions on Communications.

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

[10]  Shu-Tao Xia,et al.  LDPC Codes Based on Berlekamp-Justesen Codes with Large Stopping Distances , 2006, 2006 IEEE Information Theory Workshop - ITW '06 Chengdu.

[11]  Alon Orlitsky,et al.  Stopping set distribution of LDPC code ensembles , 2003, IEEE Transactions on Information Theory.

[12]  Emre Telatar,et al.  Finite-length analysis of low-density parity-check codes on the binary erasure channel , 2002, IEEE Trans. Inf. Theory.

[13]  Kenji Sugiyama,et al.  On the minimum weight of simple full-length array LDPC codes , 2007, ISIT.

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

[15]  Bane V. Vasic,et al.  Combinatorial constructions of low-density parity-check codes for iterative decoding , 2002, IEEE Transactions on Information Theory.

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

[17]  Tor Helleseth,et al.  On the minimum distance of array codes as LDPC codes , 2003, IEEE Trans. Inf. Theory.

[18]  Khaled A. S. Abdel-Ghaffar,et al.  Results on Parity-Check Matrices With Optimal Stopping And/Or Dead-End Set Enumerators , 2006, IEEE Transactions on Information Theory.

[19]  Alexander Vardy,et al.  On the stopping distance and the stopping redundancy of codes , 2006, IEEE Transactions on Information Theory.

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

[21]  Mario Blaum,et al.  New array codes for multiple phased burst correction , 1993, IEEE Trans. Inf. Theory.

[22]  Matthew Valenti,et al.  Turbo Codes , 2007 .