Simplified Viterbi decoding of geometrically uniform TCM codes

In this paper, we present a procedure to design maximum likelihood (MIL) decoders for the new class of geometrically uniform (GU) trellis coded modulation (TCM) codes, exploiting the algebraic properties of such codes. The proposed design has a very efficient VLSI implementation. The design of the decoders for the GUTCM codes is more complicated in comparison to the standard convolutional codes because between any pair of states in the trellis diagram of a GUTCM code, there is usually a large number of parallel transitions, and the trellis diagram of the code has a much higher degree of connectivity in comparison to binary convolutional codes. We present a novel technique for solving the parallel transitions using the algebraic structure of the GUTCM codes, which represents a significant reduction in complexity in comparison to the direct approach. The proposed technique is applied to the design of a simplified Viterbi decoder (VD) for a 64-state nonbinary GUTCM code, For this example, we obtain a 58 fold reduction in complexity for the parallel transition solver in comparison to a direct implementation.