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IEEE International Symposium on Information Theory (ISIT)

Weight distributions of LDPC code ensembles: combinatorics meets statistical physics

Changyan Di, Andrea Montanari A, Rudiger L. Urbanke

Published Dec 23, 2004• 29 citations
Publisher

Abstract

The exponent of the weight distribution of low-density parity-check (LDPC) code ensembles through a statistical physics method and a combinatorics method are computed in this paper. We show that the two approaches agree for regular LDPC codes. However, for irregular codes this is not necessarily the case.
DOI
10.1109/isit.2004.1365139

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The average weight distribution of a regular low-density parity-check (LDPC) code ensemble over a finite field is thoroughly analyzed. In particular, a precise asymptotic approximation of the average weight distribution is derived for the small-weight case, and a series of fundamental qualitative properties of the asymptotic growth rate of the average weight distribution are proved. Based on this analysis, a general result, including all previous results as special cases, is established for the minimum distance of individual codes in a regular LDPC code ensemble.
In this correspondence, the average coset weight distributions of Gallager's low-density parity-check (LDPC) code ensemble are investigated. Gallager's LDPC code ensemble consists of regular mtimesn-LDPC matrices with column weight j and row weight k. The average coset weight distribution can be derived by enumerating the number of parity-check matrices in the ensemble satisfying certain conditions. Based on combinatorial arguments, a formula for the average coset weight distribution will be proved. From the formula, we can show some properties of the average coset weight distributions such as equivalence classes of syndromes, symmetry of the distributions, and a lower bound on coset weight
The asymptotic distance distribution of regular low-density parity-check (LDPC) codes derived by Litsin and Sherlev (2002) is generalised for the case of irregular LDPC codes with constant check node degree. Numerical analysis of simple irregular LDPC codes with weight-two variable nodes shows that their minimum Hamming distance can be increased linearly with the code length when the maximum variable node degree of the code is kept below a critical value. This result is consistent with the stability condition of the LDPC codes, at least for the simple irregular case.