An upper bound on the minimum Euclidean distance for block-coded phase-shift keying

We present an upper bound on the minimum Euclidean distance d (E min) (C) for block-coded PSK. The bound is an analytic expression depending on the alphabet size q, the block length n, and the number of codewords \ C \ of the code C. The bound is valid for all block codes with q greater than or equal to 4 and with medium or high rate-codes where \ C \ > (q/3)(n). There are several well-known block codes whose d (E min) (C) is equal to our upper bound. Hence these codes are the best possible in the sense that there does not exist a code with the same q, n, and \ C \ and with a larger d (E min) (C). It also follows that for many choices of q, n, and \ C \, in particular for high rates, our upper bound on d (E min) (C) is optimal.