Improving bounds on the minimum Euclidean distance for block codes by inner distance measure optimization

The minimum Euclidean distance is a fundamental quantity for block coded phase shift keying (PSK). In this paper we improve the bounds for this quantity that are explicit functions of the alphabet size q, block length n and code size |C|. For q=8, we improve previous results by introducing a general inner distance measure allowing different shapes of a neighborhood for a codeword. By optimizing the parameters of this inner distance measure, we find sharper bounds for the outer distance measure, which is Euclidean. The proof is built upon the Elias critical sphere argument, which localizes the optimization problem to one neighborhood. We remark that any code with q=8 that fulfills the bound with equality is best possible in terms of the minimum Euclidean distance, for given parameters n and |C|. This is true for many multilevel codes.