Cryptography and Coding

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 w0 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 w0/(n−r). This work investigates high rate convolutional codes with large w. We present an explicit construction technique of free distance dfree = 4 convolutional codes with limited bitoriented trellis state complexity, high rate, and large w. The construction produces optimal codes, in the sense of maximizing w, within the classes of rate (ν(2ν−1 + 1) + 2 − 1)/(ν(2ν−1 + 2) + 2), ν ≥ 2, codes with dfree ≥ 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.