# Extremal Type II $\mathbb{Z}_4$-codes constructed from binary doubly even self-dual codes of length $40$

## TL;DR

This paper shows that all binary doubly even self-dual codes of length 40 can be derived from extremal Type II  -codes over , revealing a large number of such codes and their construction.

## Contribution

It establishes a construction method linking binary self-dual codes to extremal Type II  -codes, and quantifies the number of inequivalent codes.

## Key findings

- All binary doubly even self-dual codes of length 40 are residue codes of extremal Type II  -codes.
- At least 94,356 inequivalent extremal Type II  -codes of length 40 exist.
- The construction provides a comprehensive classification of these codes.

## Abstract

In this note, we demonstrate that every binary doubly even self-dual code of length $40$ can be realized as the residue code of some extremal Type II $\mathbb{Z}_4$-code. As a consequence, it is shown that there are at least $94356$ inequivalent extremal Type II $\mathbb{Z}_4$-codes of length $40$.