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

TL;DR

This paper introduces methods to construct many extremal Type II $\\mathbb{Z}_{2k}$-codes of various lengths, expanding the known classes of such codes with new explicit examples.

Contribution

It provides systematic construction techniques for extremal Type II $\\mathbb{Z}_{2k}$-codes from existing codes, including new codes of lengths 24, 32, 56, and 64.

Findings

Constructed extremal Type II $\\mathbb{Z}_{2k}$-codes of length 24 for $k=4,5,...,20$

Constructed extremal Type II $\\mathbb{Z}_{2k}$-codes of length 32 for $k=4,5,...,10$

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

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 $k=4,5,\ldots,20$ and extremal Type II $\mathbb{Z}_{2k}$-codes of length $32$ for $k=4,5,\ldots,10$. We also construct new extremal Type II $\mathbb{Z}_4$-codes of lengths $56$ and $64$.