For lengths
8
8
,
16
16
, and
24
24
, it is known that there is an extremal Type II
Z
2
k
\mathbb {Z}_{2k}
-code for every positive integer
k
k
. In this paper, we show that there is an extremal Type II
Z
2
k
\mathbb {Z}_{2k}
-code of lengths
32
,
40
,
48
,
56
32,40,48,56
, and
64
64
for every positive integer
k
k
. For length
72
72
, it is also shown that there is an extremal Type II
Z
4
k
\mathbb {Z}_{4k}
-code for every positive integer
k
k
with
k
≥
2
k \ge 2
.