LDPC Codes Based On The Space Of Symmetric Matrices

Let Sn(Fq) be the set of all n x n symmetric matrix over Fq. T is an undirected graph with Sn(Fq) as its vertex set and two vertices S and S’ are adjacent if and only if rank(S - S’)= 1. The vertex set of a maximal clique in the graph T corresponds to a maximal set of rank 1 of Sn(Fq). In this paper, we construct a new bipartite graph G(n,q), whose point set V(n,q)= Sn(Fq), line set L(n,q)={all the maximal set of rank 1 of Sn(Fq)}, and Let H(n, q) be the adjacent matrix of the bipartite graph G(n, q), whose rows and columns correspond to the lines and points of G(n, q). Then the null spaces over F2 of Ⅱ(n, q) and HT(n, q) (HT(n, q) is the transpose of H(n, q)) give two binary LDPC codes, denoted by C(n,q) and CT(n,q).In this paper, we determine the girth of the bipartite graph G(n,q), the minimum distance of the codes CT(n,q) and C(n,2). When the characteristic of Fq is 2, we also determine the minimum distance of C(2,q). In general, we get a lower bound of the minimum distance of C(n,q). More precisely, we obtain the following results:· The girth of the bipartite graph G(n, q) is 8;· The minimum distance of CT(n,q) equals 2q;· The minimum distance of C(2, q) equals 4q, where q is a power of 2;· The minimum distance of C(n,2) equals· The minimum distance of C(n,q) is d(C(n,q)), then d(C(n,q))≥2(?).