Midwest cousins of Barnes-Wall lattices

TL;DR

This paper introduces a method to construct new lattices called cousin lattices from rational lattices using linear transformations, with specific focus on Barnes-Wall lattices, resulting in series of unimodular lattices with notable properties.

Contribution

It provides a novel construction framework for cousin lattices, especially for Barnes-Wall lattices, including conditions for integrality, evenness, and unimodularity, and identifies new unimodular lattice series.

Findings

Constructed multi-parameter series of cousin lattices from Barnes-Wall lattices.

Identified unimodular subseries with specific ranks and high minimum norms.

Established conditions for integrality, evenness, and unimodularity of the constructed lattices.

Abstract

Given a rational lattice and suitable set of linear transformations, we construct a cousin lattice. Sufficient conditions are given for integrality, evenness and unimodularity. When the input is a Barnes-Wall lattice, we get multi-parameter series of cousins. There is a subseries consisting of unimodular lattices which have ranks $2^{d-1}\pm 2^{d-k-1}$, for odd integers $d\ge 3$ and integers $k=1,2, ..., \frac {d-1}2$. Their minimum norms are moderately high: $2^{\lfloor \frac d2 \rfloor -1}$.