Design and performance of LDPC codes extended with parity-check symbols from a larger alphabet

This paper describes a method for constructing a certain class of LDPC codes over mixed integer residue rings. These codes are essentially extended LDPC codes where the additional redundant symbols come from a larger alphabet. In particular, we address the effects that zero divisors as edge weights in the Tanner graph of LDPC codes over Z, have, from the pseudocodeword perspective. First, we show that weighting certain edges of a Tanner graph with zero divisors of Z, can reduce the number of pseudocodewords arising from its finite degree covers. Second, we prove that in adding a redundant row to a given parity-check matrix, the smaller the number of zero divisors it contains, the more effective it is in constricting the corresponding fundamental polytope. Moreover, we argue that cycles of length four are not bad provided at least one of the four corresponding edge weights is a zero divisor. The culmination of these findings is a procedure for judiciously adding redundant check nodes to the Tanner graph representation of a mixed- alphabet code such that significant performance improvements under iterative decoding may be obtained.