Many questions remain unanswered in this work. First, it is still not
clear why lexicodes, or even trellis-oriented
-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
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.