High Rate Convolutional Codes with Optimal Cycle Weights

Abstract

Consider a cycle in the state diagram of a rate (n-r)/n convolutional code. The total Hamming weight of all labels on the edges of the cycle divided by the number of edges is the average cycle weight per edge. Let w ₀ denote the minimum average cycle weight per edge over all cycles in a minimal state diagram of a convolutional code, excluding the all-zero cycle around the all-zero state. For comparison between codes of different parameters, let $w \triangleq w0/(n-r).$ This work investigates high rate convolutional codes with large w. We present an explicit construction technique of free distance d _free =4 convolutional codes with limited bit-oriented trellis state complexity, high rate, and large w. The construction produces optimal codes, in the sense of maximizing w, within the classes of rate (μ(2^μ− 1 + 1) + 2^μ− 1)/(μ(2^μ− 1 + 2) + 2^μ), μ ≥ 2, codes with d _free ≥ 3 and any code degree. An efficient exhaustive search algorithm is outlined as well. A computer search was carried out, and several codes having larger w than codes in the literature were found.