On a conjecture by Belfiore and Sol\'e on some lattices

TL;DR

This paper proves that the secrecy function reaches its maximum at y=1 for all known extremal even unimodular lattices, supporting a conjecture by Belfiore and Solé, and introduces a simple method to test the conjecture on any unimodular lattice.

Contribution

It confirms a specific case of Belfiore and Solé's conjecture for known extremal even unimodular lattices and provides a straightforward verification method for the conjecture on other lattices.

Findings

Secrecy function attains maximum at y=1 for all known extremal even unimodular lattices.

Introduces a simple method to verify or disprove the conjecture on any unimodular lattice.

Supports the conjecture by Belfiore and Solé in the special case studied.

Abstract

The point of this note is to prove that the secrecy function attains its maximum at y=1 on all known extremal even unimodular lattices. This is a special case of a conjecture by Belfiore and Sol\'e. Further, we will give a very simple method to verify or disprove the conjecture on any given unimodular lattice.