On the Minimum/Stopping Distance of Array Low-Density Parity-Check Codes

In this paper, we study the minimum/stopping distance of array low-density parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m q. In the literature, the minimum/stopping distance of these codes (denoted by d(q, m) and h(q, m), respectively) has been thoroughly studied for m 5. Both exact results, for small values of q and m, and general (i.e., independent of q) bounds have been established. For m = 6, the best known minimum distance upper bound, derived by Mittelholzer, is d(q, 6) 32. In this paper, we derive an improved upper bound of d(q, 6) 20 and a new upper bound d(q, 7) 24 by using the concept of a template support matrix of a codeword/stopping set. The bounds are tight with high probability in the sense that we have not been able to find codewords of strictly lower weight for several values of q using a minimum distance probabilistic algorithm. Finally, we provide new specific minimum/stopping distance results for m 7 and low-to-moderate values of q ≤79.

[1]  Kenji Sugiyama,et al.  On the minimum weight of simple full-length array LDPC codes , 2007, 2007 IEEE International Symposium on Information Theory.

[2]  Gwan S. Choi,et al.  FPGA based implementation of decoder for array low-density parity-check codes , 2005, Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005..

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

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

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

[6]  S. Olcer,et al.  Decoder architecture for array-code-based LDPC codes , 2003, GLOBECOM '03. IEEE Global Telecommunications Conference (IEEE Cat. No.03CH37489).

[7]  T. Mittelholzer.  Efficient encoding and minimum distance bounds of Reed-Solomon-type array codes , 2002, Proceedings IEEE International Symposium on Information Theory,.

[8]  Jing Cai,et al.  Analysis of the distribution of the number of erasures correctable by a binary linear code and the link to low-weight codewords , 2007, IET Commun..

[9]  Øyvind Ytrehus,et al.  An Efficient Algorithm to Find All Small-Size Stopping Sets of Low-Density Parity-Check Matrices , 2009, IEEE Transactions on Information Theory.

[10]  Marcel Ambroze,et al.  Addendum to “An Efficient Algorithm to Find All Small-Size Stopping Sets of Low-Density Parity-Check Matrices” , 2012, IEEE Transactions on Information Theory.

[11]  Haiyang Liu,et al.  On the Number of Minimum Stopping Sets and Minimum Codewords of Array LDPC Codes , 2010, IEEE Communications Letters.

[12]  Morteza Esmaeili,et al.  On the Stopping Distance of Array Code Parity-Check Matrices , 2009, IEEE Transactions on Information Theory.

[13]  T. Aaron Gulliver,et al.  More on the Stopping and Minimum Distances of Array Codes , 2011, IEEE Transactions on Communications.