Bounds for codes over the unit circle

Let C be a code of length n and rate R over the alphabet A(Q)=\{ \exp (2\pi ir/Q): r=O,1, \cdots ,Q-1\} , and let d(C) be the minimum Euclidean distance of C . For large n , the lower and upper bounds are obtained in parametric form on the achievable pairs (R, \delta) , where \delta = d^{2}(C)/n holds. To obtain these bounds, the arguments leading to the Gilbert bound and the Elias bound, respectively, are applied to the alphabet A(Q) . For Q \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 for \delta \leq 0.93 .