An Optimal Odd Unimodular Lattice in Dimension 72

TL;DR

The paper constructs the first known optimal odd unimodular lattice in dimension 72 by leveraging the recently discovered extremal even unimodular lattice, establishing a new link between these lattice types.

Contribution

It demonstrates a method to derive an optimal odd unimodular lattice in dimension 72 from an extremal even unimodular lattice, providing the first example of such a lattice.

Findings

Existence of an optimal odd unimodular lattice in dimension 72

Construction method from extremal even unimodular lattice

New link between even and odd unimodular lattices

Abstract

It is shown that if there is an extremal even unimodular lattice in dimension 72, then there is an optimal odd unimodular lattice in that dimension. Hence, the first example of an optimal odd unimodular lattice in dimension 72 is constructed from the extremal even unimodular lattice which has been recently found by G. Nebe.