On the Minimum Distance of Array-Based Spatially-Coupled Low-Density Parity-Check Codes

TL;DR

This paper investigates the minimum distance of array-based spatially-coupled LDPC codes, providing tight bounds for certain parameters and demonstrating how edge spreading can significantly improve minimum distance for smaller q values.

Contribution

The work derives tight upper bounds on the minimum distance for spatially-coupled array LDPC codes and shows how edge spreading enhances minimum distance for smaller q.

Findings

Tight upper bounds on minimum distance for m=3,4,5 independent of q.

Edge spreading can significantly increase minimum distance for small q.

Exhaustive search confirms improvements with careful edge spreading.

Abstract

An array low-density parity-check (LDPC) code is a quasi-cyclic LDPC code specified by two integers $q$ and $m$, where $q$ is an odd prime and $m \leq q$. The exact minimum distance, for small $q$ and $m$, has been calculated, and tight upper bounds on it for $m \leq 7$ have been derived. In this work, we study the minimum distance of the spatially-coupled version of these codes. In particular, several tight upper bounds on the optimal minimum distance for coupling length at least two and $m=3,4,5$, that are independent of $q$ and that are valid for all values of $q \geq q_0$ where $q_0$ depends on $m$, are presented. Furthermore, we show by exhaustive search that by carefully selecting the edge spreading or unwrapping procedure, the minimum distance (when $q$ is not very large) can be significantly increased, especially for $m=5$.