Abstract:
LetCbe a code of lengthnand rateRover the alphabetA(Q)=\{ \exp (2\pi ir/Q): r=O,1, \cdots ,Q-1\}, and letd(C)be the minimum Euclidean distance ofC. For largen, the lower ...Show MoreMetadata
Abstract:
LetCbe a code of lengthnand rateRover the alphabetA(Q)=\{ \exp (2\pi 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), where\delta = 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 for\delta \leq 0.93.
Published in: IEEE Transactions on Information Theory ( Volume: 32, Issue: 6, November 1986)