Extremal Type I $\mathbb{Z}_k$-codes and $k$-frames of odd unimodular lattices

TL;DR

This paper investigates the existence of $k$-frames in extremal odd unimodular lattices across various dimensions, leading to the construction of new extremal and near-extremal Type I $bZ_k$-codes for many lengths.

Contribution

It determines all integers $k$ for which extremal odd unimodular lattices contain $k$-frames, enabling the construction of new extremal and near-extremal Type I $bZ_k$-codes.

Findings

Identified all $k$-frames in extremal odd unimodular lattices in specified dimensions.

Established the existence of extremal Type I $bZ_k$-codes for multiple lengths.

Provided new constructions of near-extremal Type I $bZ_k$-codes with few exceptions.

Abstract

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