LDPC Codes Based on the Space of Symmetric Matrices over Finite Fields

TL;DR

This paper introduces a novel explicit construction method for regular LDPC codes based on symmetric matrices over finite fields, analyzing their properties including minimum and stopping distances.

Contribution

The paper presents a new construction approach for LDPC codes using symmetric matrix spaces, providing explicit classes with known girth, minimum distance, and stopping distance.

Findings

Both code classes have girth 8.

The minimum and stopping distances of one class are both 2q.

Lower bounds for minimum and stopping distances are established for the other class.

Abstract

In this paper, we present a new method for explicitly constructing regular low-density parity-check (LDPC) codes based on $\mathbb{S}_{n}(\mathbb{F}_{q})$, the space of $n\times n$ symmetric matrices over $\mathbb{F}_{q}$. Using this method, we obtain two classes of binary LDPC codes, $\cal{C}(n,q)$ and $\cal{C}^{T}(n,q)$, both of which have grith $8$. Then both the minimum distance and the stopping distance of each class are investigated. It is shown that the minimum distance and the stopping distance of $\cal{C}^{T}(n,q)$ are both $2q$. As for $\cal{C}(n,q)$, we determine the minimum distance and the stopping distance for some special cases and obtain the lower bounds for other cases.