On the Residue Codes of Extremal Type II Z4-Codes of Lengths 32 and 40

TL;DR

This paper investigates the residue codes of extremal Type II Z4-codes at lengths 32 and 40, establishing their dimensions, realizability, and uniqueness, and constructing many new codes.

Contribution

It determines the residue code dimensions for lengths 32 and 40, proves realizability of all binary doubly even self-dual codes of length 32, and constructs new extremal Type II Z4-codes.

Findings

Residue code dimensions for lengths 32 and 40 are determined.

All binary doubly even self-dual codes of length 32 are realizable as residue codes.

A unique extremal Type II Z4-code of length 32 with residue code dimension 6 is identified.

Abstract

In this paper, we determine the dimensions of the residue codes of extremal Type II Z4-codes for lengths 32 and 40. We demonstrate that every binary doubly even self-dual code of length 32 can be realized as the residue code of some extremal Type II Z4-code. It is also shown that there is a unique extremal Type II Z4-code of length 32 whose residue code has the smallest dimension 6 up to equivalence. As a consequence, many new extremal Type II Z4-codes of lengths 32 and 40 are constructed.