The asymptotic Elias upper bound of codes designed for Hamming distance is well known. Piret and Ericsson have extended this bound for codes over symmetric PSK signal sets with Euclidean distance and for codes over signal sets that form a group, with general distance function respectively. The tightness of these bounds depend on a choice of a probability distribution, and finding the distribution (optimum distribution) that leads to the tightest bound is difficult in general. In this paper we point out that these bounds are valid for codes over the wider class of distance-uniform signal sets (a signal set is referred to be distance-uniform if the Euclidean distance distribution is same from any point of the signal set). We show that optimum distributions can be found for (i) simplex signal sets, (ii) Hamming spaces and (iii) biorthogonal signal set. The classical Elias bound for arbitrary alphabet size is shown to be obtainable by specializing the extended bound to simplex signal sets with optimum distribution. We also verify Piret's conjecture for codes over 5-PSK signal set.

Geometrically Uniform (GU) codes have been a center of attention because their symmetric properties along with group algebraic structure provide benefits on their design and perfomance evaluation. We have been following a class of GU codes tha we call Strictly Geometrically Uniform (SGU) codes. Our studies had started from devising a way to get SGU trellis codes from Non-SGU (NSGU) constellations. Essentially, SGU multidimensional constellations were derived from an 1- or 2-dimensional NSGU constellations. Some simple good codes were then found, and the novelty is that they rely on symmetries of permutation of channel symbols. Applying the same method to PSK-type constellations, which is SGU, yielded again good codes, along with results regarding their algebraic structure.