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Advances in Mathematics of Communications

Upper bounds on the minimum distance of LDPC codes with double diagonal parity checks

François Arnault, Philippe Gaborit, Wouter Rozendaal, Nicolas Saussay, Gilles Zémor

Published Jul 13, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.3934/amc.2026055

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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} + \epsilon})O(nr−1r−2​+ϵ), where nnn is the code length and ϵ>0\epsilon > 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. Additionally, we show that the new upper bound also applies to the class of repeat-accumulate codes.
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We study binary quasi-cyclic LDPC codes whose parity-check matrices are full arrays of single circulant permutation matrices (CPMs), referred to here as CPM-LDPC codes. Their minimum distance is at most (J+1)!(J+1)!(J+1)!, where JJJ is the column weight. For every fixed pair of column and row weights $2\le J
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.
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In this paper, we study the minimum/stopping 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/stopping distance of these codes (denoted by d(q, m) and h(q, m), respectively) 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, is d(q, 6) 32. In this paper, we derive an improved upper bound of d(q, 6) 20 and a new upper bound d(q, 7) 24 by using the concept of a template support matrix of a codeword/stopping set. The bounds are 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/stopping distance results for m 7 and low-to-moderate values of q ≤79.