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IEEE Transactions on Information Theory

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

Yael Ben‐Haim, Simon N. Litsyn

Published Apr 28, 2006• 24 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. 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.

DOI
10.1109/tit.2006.872972

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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. 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.
We investigate the minimum distance of structured binary Low-Density Parity-Check (LDPC) codes whose parity-check matrices are of the form $ [ \mathbf{C} \, \vert \, \mathbf{M}] $ where $ \mathbf{C} $ is circulant and of column weight $ 2 $, and $ \mathbf{M} $ has fixed column weight $ r \geq 3 $ and row weight at least $ 1 $. 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(n^{\frac{r-2}{r-1} + \epsilon}) $, where $ n $ is the code length and $ \epsilon > 0 $ is arbitrarily small. This improves the previously known upper bound in $ O(n^{\frac{r-1}{r}}) $ on the minimum distance of such codes. Additionally, we show that the new upper bound also applies to the class of repeat-accumulate codes.
This correspondence studies the performance of the iterative decoding of low-density parity-check (LDPC) code ensembles that have linear typical minimum distance and stopping set size. We first obtain a lower bound on the achievable rates of these ensembles over memoryless binary-input output-symmetric channels. We improve this bound for the binary erasure channel. We also introduce a method to construct the codes meeting the lower bound for the binary erasure channel. Then, we give upper bounds on the rate of LDPC codes with linear minimum distance when their right degree distribution is fixed. We compare these bounds to the previously derived upper bounds on the rate when there is no restriction on the code ensemble.