Determining the Patterns of Low-Weight Codewords for Systematic Convolutional Codes

In this paper, we present a systematic method to determine not only the multiplicity, but also detailed patterns of low-weight parity-check components and systematic components with Hamming weight up to 4 for a rate r = 1=2 recursive systematic convolutional code, simply by referring to its generator function. The detailed patterns of the low-weight codewords are then established by identifying those with low-weight parity-check components or systematic components. This method is also applied to non-recursive systematic convolutional codes by tweaking the generator function of the recursive systematic convolutional code. The union bound is obtained by comparing it to those obtained from the transfer function and the bit error rate curve obtained from simulation results. The comparison is solely for validating the proposed method and no significant counting loss is detected. Possible applications of our proposed method to concatenated code design are briefly discussed.