LDPC Codes Based On The Space Of Alternate Matrices

Let Fq be the finite field with q elements, where q is a power of a prime. The set of all n x n alternate matrices over  is called the space of n×n alternate matrices over  and denoted by ICn(Fq). Let X1,X2 be two points in Kn(Fq), then denote lx1,x2={X1+ x(X1-X2)|x∈Fq}. Let Lk(n,q) be the set of{lX1,x2|X1, X2∈Kn(Fg), ad(X1,X2)= 1} which are called lines. Let Vk(n,q) be the set of{X|X∈Kn(Fq)} that are called points. The adjacency relation between point and line is the inclusion relation. (Vk(n, q), Lk(n, q)) constitute a bipartite graph Tk{n,q). Let Hk(n,q) be the incidence matrix of Tk(n,q), where the rows are indexed by lines and columns are indexed by points. The binary code with the parity check matrix Hk(n,q) is an LDPC code, denoted by Ck(n,q).The binary code with the parity check matrix Hk’(n, q) is also an LDPC code, denoted by Ck*(n,q).In the present paper, when q is a power of 2, we obtain the minimum distances of Ck(n,2) and Ck*(n,q). And we obtain the lower bound of the minimum distance of Ck(4, q). Prove the following theorems:Theorem A The minimum distance of Ck(n,2) is 22Theorem B Let d be the minimum distance of Ck(4, q), we have d≥ 4q4 - 2q3 + 3q2+ q+2, where q is a power of 2.Theorem C When q is the power of 2, the minimum distance of Ck*(n, q) is q+1.