Asymptotic Elias Bound for Euclidean Space Codes over Uniform Signal Sets

We extend Piret's upper bound [1] 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 [2]. The probability distribution that gives optimum bound is obtained for codes over simplex, biorthogonal signal sets and Hamming spaces