Density Evolution for GF(q) LDPC Codes Via Simplified Message-passing Sets

Abstract

A message-passing decoder for GF(q) low-density parity-check codes is defined, which uses discrete messages from a subset of all possible binary vectors of length q. The proposed algorithm is a generalization to GF(q) of Richardson and Urbanke's decoding "Algorithm E" for binary codes. Density evolution requires a mapping between the probability distribution spaces for the channel, variable and check messages, and under the proposed algorithm, exact density evolution is possible. Symmetries in the message densities permit reduction in the size of the probability distribution space. Noise thresholds are obtained for LDPC codes on discrete memoryless channels, and as with Algorithm E, are remarkably close to noise thresholds under more complex belief propagation decoding.