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In this paper, we give a new upper bound on the minimum Euclidean weight ofType II $\ZZ_{2k}$-codes and the concept of extremality for the Euclideanweights when $k=3,4,5,6$. Together with the known result, we demonstrate thatthere is an extremal Type II $\ZZ_{2k}$-code of length $8m$ $(m \le 8)$ when$k=3,4,5,6$.

An Upper Bound on the Minimum Weight of Type II $\ZZ_{2k}$-Codes