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
.