Rank 72 high minimum norm lattices

TL;DR

This paper introduces a family of unimodular lattices derived from polarizations, identifying rank 72 lattices with the highest known minimum norm of 6, and explores possibilities for higher norms.

Contribution

It defines a new family of lattices L(M,N,k) and establishes rank 72 lattices with the highest known minimum norm of 6, advancing lattice theory.

Findings

Minimum norms 4, 6, 8 observed in lattices

Rank 72 lattices with norm 6 are the highest known

Some lattices in higher dimensions have moderately high norms

Abstract

Given a polarization of an even unimodular lattice and integer $k\ge 1$, we define a family of unimodular lattices $L(M,N,k)$. Of special interest are certain $L(M,N,3)$ of rank 72. Their minimum norms lie in $\{4, 6, 8\}$. Norms 4 and 6 do occur. Consequently, 6 becomes the highest known minimum norm for rank 72 even unimodular lattices. We discuss how norm 8 might occur for such a $L(M,N,3)$. We note a few $L(M,N,k)$ in dimensions 96, 120 and 128 with moderately high minimum norms.