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

摘要

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