Low Density Parity Check Codes Based on Finite Geometries and Balanced Incomplete Block Design

Low Density Parity Check (LDPC) Codes are the class of linear block codes which provide near capacity performance on large collection of data transmission channels while simultaneously feasible for implementable decoders. LDPC codes were first proposed by Gallger in 1967 [1] and were rarely used until their rediscovery by Mackay, Luby and others [9][10][11]. Much research is devoted to characterize LDPC codes and enhancing their performance on different channels. Tanner formulated a bipartite graph representation of low density codes now known as Tanner graphs [12]. We can associate a graph T to the LDPC code using basic graph theory. Let T = {(V,ε )}, with V being a set of vertices or nodes V and ε is a set of edges E connecting the vertices. A cycle of a graph T, sometimes also called a circuit, is a subset of the edge set E of T that forms a path such that the first node of the path corresponds to the last. The length of cycle is the number of edges associated with it. The girth of a graph is the length of shortest cycle. A bipartite graph is one in which the nodes can be partitioned into two disjoint classes, and . An edge of the graph may connect a node of one class to a node of the other class , but there are no edges connecting nodes of the same class [12]. In Tanner graphs, one class of nodes of bipartite graph is associated to bits whereas other class is associated with check equations. Although tanner graphs are a good visual tool for studying the decoding algorithms for LDPC codes, they cannot be regarded as good design tools. 1 V 2 V