We examine the theoretical underpinnings of lexicodes and investigate various generalizations. Our intent is to graft desired decoding properties onto the heuristically good features of lexicodes. In this way, we develop a means for designing good codes for specific tasks. For example, we can fashion good codes which maintain a desired dimension, error-correction capability, and memory constraint.
Section 2 opens with a definition of
the lexicographic construction, which iteratively
constructs generator matrices for the family of
minimum distance d lexicodes, for any d > 0.
The lexicographic construction is, in turn, a special case
of the generalized lexicographic construction, which we abbreviate
as the
-construction. This generalized construction
encompasses a variety of code families which graft different
properties onto lexicode features.
Section 3 is devoted to various instances of the
-construction. Specifically, Section 3.1 analyzes the role of
lexicodes as one of these instances and describes tools for
efficiently constructing them. In Section 3.2 we
demonstrate that a simple modification of the lexicode generator
produces ``trellis-oriented'' lexicodes that locally minimize trellis
complexity. These codes exhibit the same ``approximately optimal''
features of lexicodes but often have a much lower trellis decoding
complexity, in some cases reaching the lowest trellis complexity known
for their length, rate, and minimum distance. This modification
provides a natural heuristic for transforming a given code
into a similar code with (often) a better trellis complexity. Finally,
in Section 3.3 we examine
methods for designing codes with various trellis characteristics, such
as constrained state complexity or constrained Viterbi decoding complexity.
These methods might be particularly useful
for VLSI-based decoding,
where implementation constraints favor a small trellis state complexity, and for
mobile communications, where portability severely limits power
consumption.
Section 4 investigates the coset relationship which
supports the
-construction. This relationship establishes bounds on
the parameters of the codes produced by the construction and suggests
a natural algorithm for computing these codes. The bounds
established, though loose, are asymptotically tighter than the lexicode
bounds of Brualdi and Pless [3]. We also use the coset
relationship to bound the computation time of the
-construction
in Section 5.
Finally, in Section 6 we discuss various applications
of the
-construction and present our conclusions.
The appendix lists simulation results for the some of the algorithms
described in the paper, extending the list of lexicode parameters
originally published in [5].