Future directions

Many questions remain unanswered in this work. First, it is still not clear why lexicodes, or even trellis-oriented $ \mathfrak{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. Furthermore, one should note that the exponential time and space bound (with respect to co-dimension) of Algorithm 1 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 1 would allow for the design of lower rate codes with good trellis decoding properties. Finally, the iterated method in which lexicodes are constructed is reminiscent of concatenated codes, and the connection to concatenated code design deserves further attention.