Construction of $s$-extremal optimal unimodular lattices in dimension 52

TL;DR

This paper reports the first construction of an $s$-extremal optimal unimodular lattice in dimension 52 using a self-dual $ ext{F}_5$-code, and also constructs two neighboring lattices with similar properties.

Contribution

It introduces a novel construction method for $s$-extremal optimal unimodular lattices in dimension 52 via self-dual $ ext{F}_5$-codes and explores neighboring lattices.

Findings

First $s$-extremal optimal unimodular lattice in dimension 52 constructed.

Two neighboring $s$-extremal optimal lattices also constructed.

Lattice constructed from a self-dual $ ext{F}_5$-code via Construction A.

Abstract

An $s$-extremal optimal unimodular lattice in dimension $52$ is constructed for the first time. This lattice is constructed from a certain self-dual $\mathbb{F}_5$-code by Construction A. In addition, as neighbors of the lattice, two more $s$-extremal optimal unimodular lattices are constructed.