Upper Bounds on the Minimum Distance of Structured LDPC Codes

TL;DR

This paper establishes tighter upper bounds on the minimum distance of structured LDPC codes with specific matrix forms, demonstrating that their minimum distance grows slower than previously believed, which impacts code design and analysis.

Contribution

The paper provides a new upper bound on the minimum distance of structured LDPC codes, improving upon prior bounds and offering insights into their fundamental limitations.

Findings

Minimum distance is in O(n^{(r-2)/(r-1)+ε})

Improves previous upper bound of O(n^{(r-1)/r})

Impacts code design by clarifying limitations of structured LDPC codes

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.