Construction of extremal Type II $\mathbb{Z}_{8}$-codes via doubling method

TL;DR

This paper introduces a doubling method to construct extremal Type II $ ext{Z}_8$-codes, leading to the discovery of new codes with optimal binary residue codes, advancing the classification of these mathematical objects.

Contribution

A novel doubling technique for constructing extremal Type II $ ext{Z}_8$-codes from known codes, enabling the creation of previously unknown codes of length 32.

Findings

Constructed at least ten new extremal Type II $ ext{Z}_8$-codes of length 32.

Developed an algorithm for code construction based on the doubling method.

Binary residue codes of the new codes are optimal $[32,15]$ binary codes.

Abstract

Extremal Type II $\mathbb{Z}_{8}$-codes are a class of self-dual $\mathbb{Z}_{8}$-codes with Euclidean weights divisible by $16$ and the largest possible minimum Euclidean weight for a given length. We introduce a doubling method for constructing a Type II $\mathbb{Z}_{2k}$-code of length $n$ from a known Type II $\mathbb{Z}_{2k}$-code of length $n$. Based on this method, we develop an algorithm to construct new extremal Type II $\mathbb{Z}_8$-codes starting from an extremal Type II $\mathbb{Z}_8$-code of type $(\frac{n}{2},0,0)$ with an extremal $\mathbb{Z}_4$-residue code and length $24, 32$ or $40$. We construct at least ten new extremal Type II $\mathbb{Z}_8$-codes of length $32$ and type $(15,1,1)$. Extremal Type II $\mathbb{Z}_8$-codes of length $32$ of this type were not known before. Moreover, the binary residue codes of the constructed extremal $\mathbb{Z}_8$-codes are optimal $[32,15]$ binary codes.