On the Existence of Frames of Some Extremal Odd Unimodular Lattices and Self-Dual Zk-Codes

TL;DR

This paper investigates the existence of frames in certain extremal odd unimodular lattices across multiple dimensions, establishing links to the existence of specific self-dual Zk-codes and providing new code constructions.

Contribution

It determines all positive integers k for which these lattices contain a k-frame, leading to new existence results for extremal and near-extremal Type I Zk-codes.

Findings

Existence of k-frames in extremal lattices for specific dimensions.

Construction of extremal Type I Zk-codes for various lengths.

Identification of a near-extremal Zk-code of length 28.

Abstract

For some extremal (optimal) odd unimodular lattices L in dimensions n=12,16,20,32,36,40 and 44, we determine all positive integers k such that L contains a k-frame. This result yields the existence of an extremal Type I Zk-code of lengths 12,16,20,32,36,40 and 44 and a near-extremal Type I Zk-code of length 28 for positive integers k with only a few exceptions.