On lattices of maximal index two

TL;DR

This paper investigates the properties of Euclidean lattices with maximal index two, establishing an upper bound on the dimension of perfect lattices not derived from cross-sections.

Contribution

It proves that perfect lattices of maximal index two, not obtained via cross-sections, have dimension at most 5, providing new bounds in lattice theory.

Findings

Perfect lattices of maximal index two have dimension ≤ 5 if not from a cross-section.

The paper establishes bounds on lattice dimensions based on index and perfection.

Provides theoretical limits for lattice structures in Euclidean spaces.

Abstract

The maximal index of a Euclidean lattice L of dimension n is the maximal index of the sub-lattices of L spanned by n independent minimal vectors of L. In this paper, we prove that a perfect lattice of maximal index two not provided by a cross-section has dimension at most 5.