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IEEE International Symposium on Information Theory (ISIT)

Upper bounds on the rate of LDPC codes as a function of minimum distance

Yael Ben‐Haim, Simon N. Litsyn

Published Jan 1, 2005• 3 citations
Publisher

Abstract

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. These bounds are based on combinatorial arguments and linear programming. They improve on the previous bounds due to Burshtein et al. It is proved that at least for high rate LDPC codes have worse relative minimum distance than the one guaranteed by the Gilbert-Varshamov bound

DOI
10.1109/isit.2005.1523290

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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. These bounds are based on combinatorial arguments and linear programming. They improve on the previous bounds due to Burshtein et al. It is proved that at least for high rate LDPC codes have worse relative minimum distance than the one guaranteed by the Gilbert-Varshamov bound
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.
We investigate the minimum distance of structured binary Low-Density Parity-Check (LDPC) codes whose parity-check matrices are of the form [C∣M][\mathbf{C} \vert \mathbf{M}][C∣M] where C\mathbf{C}C is circulant and of column weight 222, and M\mathbf{M}M has fixed column weight r≥3r \geq 3r≥3 and row weight at least 111. These codes are of interest because they are LDPC codes which come with a natural linear-time encoding algorithm. We show that the minimum distance of these codes is in O(nr−2r−1+ε)O(n^{\frac{r-2}{r-1} + ε})O(nr−1r−2​+ε), where nnn is the code length and ε>0ε> 0ε>0 is arbitrarily small. This improves the previously known upper bound in O(nr−1r)O(n^{\frac{r-1}{r}})O(nrr−1​) on the minimum distance of such codes.
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.