Linear-Congruence Constructions of Low-Density Parity-Check Codes

Abstract

Low-Density Parity-Check codes (LDPCC) with Iterative Belief Propagation (Message Passing) decoding are attractive alternatives to Turbo codes. LDPCC previously discussed in the literature have involved matrices constructed using random techniques. In this paper, we discuss construction techniques for LDPCC involving multiple permutation matrices, each specified by a linear congruence. Construction options depend on the size of the parity-check matrix and the rate of the code. We relate desirable properties of the code to the parameters defining the linear congruences specifying the permutation matrices used to construct the code. For example, codes with few or no 4-cycles can be readily constructed. We summarize the construction options and describe selection processes for the parameters of the congruences. We then provide performance results for regular parity-check matrices constructed by random and the linear-congruence techniques for rate 1/2 transmit block-size 980 and rate 4/7 transmit block-size 847 codes. We introduce a symmetric channel model for decoding with the iterative belief propagation algorithm and describe its use as a heuristic for deciding whether a code is likely better or worse than most codes of the given rate and block size.