The design of a trellis code involves selecting a labeled constellation and a convolutional code to maximize certain design criteria. This talk addresses the choice of constellation labeling for trellis codes designed for additive white Gaussian noise (AWGN) or for fading channels (such as those found in many wireless environments).

For AWGN channels, the most important criterion is the code Euclidean distance (CED). For channels with uncorrelated Rayleigh fading, the effective code length (ECL) and code product distance (CPD) are the design criteria of choice. For channels with correlated fading, codes that maximize the periodic effective code length (PECL) and code periodic product distance (CPPD) provide good performance when combined with periodic interleaving.

The set-partitioning labeling strategy proposed by Ungerboeck has produced excellent trellis codes for AWGN channels. Set partitioning simplifies the code design problem by allowing for one or more uncoded bits. However, for channels with fading, good performance is possible only when there are no uncoded bits. There is no clear advantage to using set partitioning to label constellations for fading-channel trellis codes.

The central question regarding constellation labeling is whether a particular labeling will lead to a trellis code that maximizes the design criteria for the specified complexity. Simply put, can one be sure that a different labeling won't produce a better code? We resolve this question by demonstrating that certain labelings are guaranteed to produce the trellis codes that maximize the design criteria listed above.

The same labelings produce the trellis codes that maximize CED, CPD, and CPPD. These constellations, called distance-superior constellations, produce the best trellis codes for both AWGN and fading channels. (The ECL and PECL criteria are independent of constellation labeling.) In general, a different convolutional code is required to maximize each of the three metrics, but the same distance superior constellation can be used in each case.

We also demonstrate that in general neither set partitioning nor Gray labeling is necessary or sufficient for producing a labeling that is distance superior. An example is given of a distance inferior 16-QAM set partitioning and a distance inferior 16-PSK gray labeling.