Asymptotic Elias bound for Euclidean space codes over uniform signal sets

We extend Piret's (1986) upper bound to codes over uniform signal sets (a signal set is referred to be uniform if the Euclidean distance distribution is same from any point in the signal set) which include as a special case codes over symmetric PSK signal sets and all signal sets matched to groups (Loeliger 1991). The probability distribution that gives optimum bound is obtained for codes over simplex, biorthogonal signal sets and Hamming spaces.