Self-dual codes over $\mathbb{F}_5$ and $s$-extremal unimodular lattices

TL;DR

This paper constructs new s-extremal unimodular lattices in multiple dimensions using self-dual codes over 5, introducing codes with larger minimum weights than previously known, and also derives related codes with improved parameters.

Contribution

It presents the first construction of s-extremal unimodular lattices in these dimensions via self-dual 5-codes, and introduces new codes with larger minimum weights.

Findings

Constructed new s-extremal unimodular lattices in dimensions 38, 40, 42, 44.

Obtained a self-dual [44,22,14] code over 5.

Derived a [43,22,13] code over 5 with improved minimum weight.

Abstract

New $s$-extremal extremal unimodular lattices in dimensions $38$, $40$, $42$ and $44$ are constructed from self-dual codes over $\mathbb{F}_5$ by Construction A. In the process of constructing these codes, we obtain a self-dual $[44,22,14]$ code over $\mathbb{F}_5$. In addition, the code implies a $[43,22,13]$ code over $\mathbb{F}_5$. These codes have larger minimum weights than the previously known $[44,22]$ codes and $[43,22]$ codes, respectively.