Extremal Type II ${\mathbb{Z}}_4$-codes of lengths 56 and 64

Abstract

Type II ${\mathbb{Z}}_4$-codes are a remarkable class of self-dual ${\mathbb{Z}}_4$-codes. A Type II ${\mathbb{Z}}_4$-code of length n exists if and only if n is divisible by eight. For lengths up to 48, extremal Type II ${\mathbb{Z}}_4$-codes are known. In this note, extremal Type II ${\mathbb{Z}}_4$-codes of lengths 56 and 64 are constructed for the first time.