Local coordinates of a trellis source code

Let G_k be the de Bruijn graph with 2^k vertices. When reproduction labels from a 4-letter alphabet A₄ are assigned linearly to the edges of G_k using a three row binary generating matrix, a 2^k-state trellis code results which is suitable for encoding a source with alphabet A₄ at a rate of one bit per source sample with respect to the Hamming fidelity criterion. There is a special type of code which results in this way called systematic code. A systematic code belongs to three sets of codes, each set isomorphic to a residue class ring of polynomials over GF(2). A systematic code sweeps out a cycle in each of its three residue class rings via repeated multiplication by the residue class of polynomial x. The local coordinates of a systematic code are non-negative integer parameters which describe how the code is positioned in each of its three cycles. Our main result is that a systematic code is uniquely determined by its local coordinates, except in rare instances. The paper concludes with some preliminary work indicating how a code with good distortion performance in coding a memoryless source with alphabet A₄ can be found in relatively small sets of systematic codes whose local coordinates are of a certain type.