On Asymptotic Elias Bound for Euclidean Space Codes over Distance-Uniform Signal Sets

The asymptotic Elias upper bound of codes designed for Hamming distance is well known. Piret [3] and Ericsson [4] 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 a general distance function respectively. The tightness of these bounds depend on a choice of a probability distribution, and finding the distribution (called optimum distribution henceforth) that leads to the tightest bound is difflcult in general. In [l] these bounds were extended for codes over the wider class of distance-uniform signal sets. In this paper we obtain optimum distributions for codes over signal sets matched [2] to (i) dihedral group, (ii) dicyclic group (iii) binary tetrahedral group (iv) binary octahedral group (v) binary icosahedral group and (vi) n-dimensional cube. F'urther we compare the bounds of codes over these signal sets based on the spectral rate.