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arXiv:2501.19125

Upper Bounds on the Minimum Distance of Structured LDPC Codes

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

Published Jan 31, 2025• 0 citations
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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 $O(n^{\frac{r-2}{r-1} + ε})$, where $n$ is the code length and $ε> 0$ is arbitrarily small. This improves the previously known upper bound in $O(n^{\frac{r-1}{r}})$ on the minimum distance of such codes.

DOI
10.48550/arxiv.2501.19125

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