Advances in Optimal Structured Source Code Design

A memoryless equiprobable source with four-letter alphabet is to be trellis encoded at an encoding rate of one code bit per source sample with respect to the Hamming fidelity criterion. Each possible trellis code for this encoding task is characterized by a binary generating matrix of fixed dimensions via which reproduction labels from the source alphabet are linearly assigned to the trellis edges. The selected code is required to be structured in the sense that the irreducible factors of the generating matrix rows are binary primitive polynomials of pre-specified degrees and multiplicities, and the code must be optimal in the sense that the time-averaged expected Hamming reproduction error per source sample is minimized subject to the structural constraint. Methodology is presented to assist in solving this optimal structured code design problem, based upon an embedding of a structured set of trellis codes within an abelian group of odd order.