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

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, $\mathcal{C}(n,q)$ and ${\mathcal{C}}^{\mathcal{T}}(n,q)$, 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 ${\mathcal{C}}^{\mathcal{T}}(n,q)$ are both $2q$. As for $\mathcal{C}(n,q)$, we determine the minimum distance and the stopping distance for some special cases and obtain the lower bounds for other cases.