On the Existence of Extremal Type II Z2k-Codes

TL;DR

This paper proves the existence of extremal Type II Z2k-codes for various lengths and parameters, extending known results and establishing new existence results for lengths up to 72.

Contribution

It demonstrates the existence of extremal Type II Z2k-codes for lengths 32 to 64 for all positive integers k, and for length 72 with specific parameters, advancing coding theory knowledge.

Findings

Existence of extremal Type II Z2k-codes for lengths 32, 40, 48, 56, 64 for all k.

Existence of extremal Type II Z4k-codes for length 72 and k ≥ 2.

Extension of known extremal code existence results to new lengths and parameters.

Abstract

For lengths 8,16 and 24, it is known that there is an extremal Type II Z2k-code for every positive integer k. In this paper, we show that there is an extremal Type II Z2k-code of lengths 32,40,48,56 and 64 for every positive integer k. For length 72, it is also shown that there is an extremal Type II Z4k-code for every positive integer k with k \ge 2.