An upper bound on the minimum weight of Type II Z2k-codes

In this paper, we give a new upper bound on the minimum Euclidean weight of Type II Z2k-codes and the concept of extremality for the Euclidean weights when k=3,4,5,6. Together with the known result, we demonstrate that there is an extremal Type II Z2k-code of length 8m (m⩽8) when k=3,4,5,6.