Future directions

Many questions remain unanswered in this work. First, it is still not clear why lexicodes, or even trellis-oriented G-codes, have such good code parameters. Second, we suspect that the bounds on parameters of lexicodes can be improved by a more sophisticated count of the worst-case companion pairings. Finally, one should note that the exponential time and space bound (with respect to co-dimension) of Algorithm 2.12 may be improved by various approximation techniques. The actual continuation of the algorithm depends only on the properties of a few coset leaders, and a heuristic approach to choosing these leaders might work well. Speeding up Algorithm 2.12 would allow for the design of lower rate codes with good trellis decoding properties.