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

In this paper, we give a new upper bound on the minimum Euclidean weight of Type II Z 2 k -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 Z 2 k -code of length 8m ( m ⩽ 8 ) when k = 3 , 4 , 5 , 6 .