Upper Bounds on the Minimum Distance of Structured LDPC Codes

TL;DR

This paper investigates the minimum distance of structured binary LDPC codes with parity-check matrix $\mathbf{H}=[\mathbf{C} \mid \mathbf{M}]$, where $\mathbf{C}$ is circulant with column weight $2$ and $\mathbf{M}$ has column weight $r \ge 3$, preserving a linear-time encoding mechanism. It introduces a quasi-collision packing framework that converts sums of $\mathbf{M}$-columns into controlled sums of circulant columns, enabling a bound on the existence of low-weight nonzero codewords. For fixed $r \ge 3$ and large blocklength $n$, it proves $d_{\min} = O(n^{\frac{r-2}{r-1} + \epsilon})$ for any $\epsilon>0$, with $\epsilon$ arbitrarily small via parameter tuning. This tightens the prior upper bound $O(n^{(r-1)/r})$, narrows the distance-gap for these codes, and supports efficient encoding and security considerations in applications such as multiparty computation.

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.

Theorems & Definitions (6)

• Theorem 1
• Proposition 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6