Abstract
For lengths 8,16 and 24, it is known that there is an extremal Type II Z2k-code for every positive integer k. In this paper, we show that there is an extremal Type II Z2k-code of lengths 32,40,48,56 and 64 for every positive integer k. For length 72, it is also shown that there is an extremal Type II Z4k-code for every positive integer k with k \ge 2.
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