Bounds for codes over the unit circle

Let be a code of length and rate over the alphabet , and let be the minimum Euclidean distance of . For large , the lower and upper bounds are obtained in parametric form on the achievable pairs , where holds. To obtain these bounds, the arguments leading to the Gilbert bound and the Elias bound, respectively, are applied to the alphabet . For , 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 .