  An asymptotically optimal trellis-coded modulation (TCM) encoder requires the
joint design of the encoder and the binary labeling of the constellation. Since
analytical approaches are unknown, the only available solution is to perform an
exhaustive search over the encoder and the labeling. For large constellation
sizes and/or many encoder states, however, an exhaustive search is unfeasible.
Traditional TCM designs overcome this problem by using a labeling that follows
the set-partitioning principle and by performing an exhaustive search over the
encoders. In this paper we study binary labelings for TCM and show how they can
be grouped into classes, which considerably reduces the search space in a joint
design. For 8-ary constellations, the number of different binary labelings that
must be tested is reduced from 8!=40320 to 240. For the particular case of an
8-ary pulse amplitude modulation constellation, this number is further reduced
to 120 and for 8-ary phase shift keying to only 30. An algorithm to generate
one labeling in each class is also introduced. Asymptotically optimal TCM
encoders are tabulated which are up to 0.3 dB better than the previously best
known encoders.
