On maximum length convolutional codes under a trellis complexity constraint

We look at convolutional codes with maximum possible code length for prescribed redundancy, conditioned on constraints on the free distance and on the bit-oriented trellis state complexity. Rate (n−1)/n codes have been studied to some extent in the literature, but more general rates have not been studied much. In this work we consider convolutional codes of rate (n−r)/n, r⩾1. Explicit construction techniques for free distance d_free=3 and 4 codes are described. For codes with r=2, an efficient exhaustive search algorithm is outlined. For the more general case with r⩾2, a heuristic approach is suggested. Several new codes were found for r=1 and in particular for r=2 and 3. Compared to previously known codes of similar free distance and complexity constraints, the new codes have either strictly higher rate, or the same rate but smaller low distance multiplicities.