Embedding theory of lattices and its application for $2$-integrable lattices

TL;DR

This paper studies $s$-integrable lattices, introduces a new embedding method into unimodular lattices, and identifies minimal non 2-integrable lattices with specific ranks and determinants.

Contribution

It presents a novel embedding technique for lattices into unimodular lattices and discovers new minimal non 2-integrable lattices of rank 12 with determinant 15.

Findings

Identified two new minimal non 2-integrable lattices with determinant 15.

Developed an embedding method into unimodular lattices for lattice analysis.

Confirmed minimality of previously known non 2-integrable lattices.

Abstract

For a positive integer $s$, a lattice $L$ is said to be $s$-integrable if $\sqrt{s}\cdot L$ is isometric to a sublattice of $\mathbb{Z}^n$ for some integer $n$. Conway and Sloane found two minimal non $2$-integrable lattices of rank $12$ and determinant $7$ in 1989. We find two more ones of rank $12$ and determinant $15$. Then we introduce a method of embedding a given lattice into a unimodular lattice, which plays a key role in proving minimality of non $2$-integrable lattices and finding candidates for non $2$-integrable lattices.