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Problems of Information Transmission

Upper bound on the minimum distance of LDPC codes over GF(q) based on counting the number of syndromes

Alexey Frolov

Published Jan 1, 2016• 1 citation
Publisher

Abstract

In [1] a syndrome counting based upper bound on the minimum distance of regular binary LDPC codes is given. In this paper we extend the bound to the case of irregular and generalized LDPC codes over GF ( q ). A comparison with the lower bound for LDPC codes over GF ( q ), upper bound for the codes over GF ( q ), and the shortening upper bound for LDPC codes is made. The new bound is shown to lie under the Gilbert–Varshamov bound at high rates.

DOI
10.1134/s0032946016010026

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In [1] a syndrome counting based upper bound on the minimum distance of regular binary LDPC codes is given. In this paper we extend the bound to the case of irregular and generalized LDPC codes over GF ( q ). A comparison with the lower bound for LDPC codes over GF ( q ), upper bound for the codes over GF ( q ), and the shortening upper bound for LDPC codes is made. The new bound is shown to lie under the Gilbert–Varshamov bound at high rates.
New upper bounds on the rate of low-density parity-check (LDPC) codes as a function of the minimum distance of the code are derived. The bounds apply to regular LDPC codes, and sometimes also to right-regular LDPC codes. Their derivation is based on combinatorial arguments and linear programming. The new bounds improve upon the previous bounds due to Burshtein et al. It is proved that at least for high rates, regular LDPC codes with full-rank parity-check matrices have worse relative minimum distance than the one guaranteed by the Gilbert-Varshamov bound.
In this work, we present an upper bound on the minimum 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 distance of these codes (denoted by d(q,m)) 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 (IEEE Int. Symp. Inf. Theory, Jun./Jul. 2002), is d(q, 6) ≤ 32. In this work, we derive an improved upper bound of d(q, 6) ≤ 20 by using the concept of a template support matrix of a codeword. The bound is 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 distance results for m ≤ 6 and low-to-moderate values of q ≤ 79.