The two parameters commonly used for comparing TCM schemes are the free Euclidean distance and the error probability. Since designs of TCM schemes are typically based on a search among a wide subclass, it is extremely important that computationally efficient algorithms for the computation of both free distance and error probability be available. Essentially, these algorithms can be classified into two categories. The first one, based on consideration of pairs of states, is extremely general, but has a computational complexity increasing with N2 (where N is the number of states in the trellis scheme). The second category includes algorithms whose computational complexity increases with N, but they are applicable only to special classes of codes. Interest in TCM schemes with a large number of states fosters the investigation of sufficient conditions for the performance of a specific TCM scheme to be evaluated using algorithms of the second category. If this is possible for the evaluation of free distance, we say that the scheme has the Uniform Distance Property (UDP), while if this is possible for the evaluation of an upper bound (based on union-Bhat-tacharyya or union-Chernoff bound) to error probability, we say that it has the Uniform Error Property (UEP). In this paper, we derive a set of conditions that are sufficient for UDP and UEP. We define the class of uniform TCM schemes, and we prove that they have both UDP and UEP. Uniformity depends on the metric properties of the two subconstellations resulting from the first step in set partitioning, as well as on the assignment of binary labels to channel symbols (the latter condition was disregarded in previous works on the subject). These concepts and results are also extended to encompass transmission over a (not necessarily Gaussian) memoryless channel where the metric used for detection may not be maximum-likelihood. An appropriate distance measure is defined, which generalizes the Euclidean distance. Then it is proved that uniformity of a TCM scheme can also be defined under this new distance. In particular, our results hold for channels with phase offset or independent, amplitude-only fading. Examples are included to illustrate the applicability of our results.
