Bounds for codes over the unit circle

LetCbe a code of lengthnand rateRover the alphabetA(Q)={ exp (2pi ir/Q): r=O,1, cdots ,Q-1}, and letd(C)be the minimum Euclidean distance ofC. For largen, the lower and upper bounds are obtained in parametric form on the achievable pairs(R, delta), wheredelta = d^{2}(C)/nholds. To obtain these bounds, the arguments leading to the Gilbert bound and the Elias bound, respectively, are applied to the alphabetA(Q). ForQ rightarrow infty, they are shown to be expressible in terms of the modified Bessel function of the first kind. The Elias type bound is compared with the Kabatyanskii-Levenshtein (K-L) bound that holds for less restrictive alphabets. It turns out that our upper bound improves the K-L bound fordelta leq 0.93.