Extremal Type II -codes of lengths 56 and 64

Type II Z 4 -codes are a remarkable class of self-dual Z 4 -codes. A Type II Z 4 -code of length n exists if and only if n is divisible by eight. For lengths up to 48, extremal Type II Z 4 -codes are known. In this note, extremal Type II Z 4 -codes of lengths 56 and 64 are constructed for the first time.