Construction of extremal Type II ${\mathbb{Z}}_{2k}$-codes

Abstract

We give methods for constructing many self-dual ${\mathbb{Z}}_m$-codes and Type II ${\mathbb{Z}}_{2k}$-codes of length $2n$ starting from a given self-dual ${\mathbb{Z}}_m$-code and Type II ${\mathbb{Z}}_{2k}$-code of length $2n$, respectively. As an application, we construct extremal Type II ${\mathbb{Z}}_{2k}$-codes of length $24$ for $k=4,5,\dots ,20$ and extremal Type II ${\mathbb{Z}}_{2k}$-codes of length $32$ for $k=4,5,\dots ,10$. We also construct new extremal Type II ${\mathbb{Z}}_4$-codes of lengths $56$ and $64$.

25 pages