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 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 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 difficult in general. In B. Sundar Rajan, et al. 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 (H.A. Loeliger, 1991) to (i) dihedral group, (ii) dicyclic group, (iii) binary tetrahedral group, (iv) binary octahedral group, (v) binary icosahedral group and (vi) n-dimensional cube. Further we compare the bounds of codes over these signal sets based on the spectral rate.