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
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.