Searching for high-rate convolutional codes via binary syndrome trellises

Rate R = (c-1)/c convolutional codes of constraint length ¿ can be represented by conventional syndrome trellises with a state complexity of s = ¿ or by binary syndrome trellises with a state complexity of s = ¿ or s = ¿ + 1, which corresponds to at most 2^s states at each trellis level. It is shown that if the parity-check polynomials fulfill certain conditions, there exist binary syndrome trellises with optimum state complexity s = ¿. The BEAST is modified to handle parity-check matrices and used to generate code tables for optimum free distance rate R = (c - 1)=c, c = 3; 4; 5, convolutional codes for conventional syndrome trellises and binary syndrome trellises with optimum state complexity. These results show that the loss in distance properties due to the optimum state complexity restriction for binary trellises is typically negligible.