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IEEE International Symposium on Information Theory (ISIT)

An upper bound on the minimum distance of LDPC codes over GF(q)

Alexey Frolov

Published Jun 1, 2015• 9 citations
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). The comparison to the lower bound for LDPC codes over GF(q) and to the upper bound for non-binary codes is done. The new bound is shown to lie under the Gilbert-Varshamov bound at high rates.

DOI
10.1109/isit.2015.7282984

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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). The comparison to the lower bound for LDPC codes over GF(q) and to the upper bound for non-binary codes is done. 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 paper, we study the asymptotic minimum distance of non-binary cluster-LDPC codes whose subjacent binary parity-check matrix is composed of localized density of ones, concentrated in clusters of bits. A particular attention is given to cluster codes represented by ultra-sparse bipartite graphs, in the sense that each symbol-node is connected to exactly dv= 2 constraint-nodes. We derive a lower bound on the minimum distance of non-binary cluster-LDPC codes and we show that there exist ensembles of ultra-sparse codes whose minimum distance grows linearly with the code length (with probability going to 1 as the code length goes to infinity). This result is in contrast with “classical” non-binary LDPC codes based on graphs with strictly regular dv= 2 symbol-nodes, whose minimum distance grows at most logarithmically with the code length. We also show that one can build practical non-binary cluster-LDPC codes with various finite codeword lengths, whose minimum distance is close to the Gilbert-Varshamov bound.