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_Emin(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⩾4 and with medium or high rate-codes where |C|>(q/3) ⁿ. There are several well-known block codes whose d_Emin (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_Emin(C). It also follows that for many choices of q, n, and |C|, in particular for high rates, our upper bound on d_Emin(C) is optimal