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

TL;DR

This paper generalizes the doubling method to Type II $\mathbb{Z}_{2k}$-codes and extends it to $\mathbb{Z}_{2^m}$-codes, enabling systematic construction of extremal codes. Focusing on $m=3$ and lengths $24$, $32$, and $40$, the authors devise an algorithm that starts from an extremal code with a maximal $\mathbb{Z}_4$-residue and produces extremal $\mathbb{Z}_8$-codes of length $n$ with type $(\frac{n}{2}-1,1,1)$. In particular, they obtain at least ten inequivalent extremal $\mathbb{Z}_8$-codes of length $32$ and type $(15,1,1)$, with binary residue codes that are optimal $[32,15]$ codes, and classify these into ten high-signal weight-distribution classes. The work combines theoretical doubling constructions with a practical algorithm (Algorithm C) and computational verification to expand the catalog of extremal $\mathbb{Z}_8$-codes and their residues.

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.

Theorems & Definitions (14)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof