The Belfiore-Sole conjecture and a certain technique for verifying it for a given lattice

Abstract

We discuss a technique provided by Ernvall-Hytonen for verifying the Belfiore-Sole conjecture for unimodular lattices, a conjecture that arises in the theory of wiretap lattice codes for the Gaussian channel. We provide an alternative proof of a key lemma of Ernvall-Hytonen that avoids dependence on a machine for verification, and as a further example of the technique, we verify the Belfiore-Sole conjecture for unimodular lattices in dimension 34 that arise from certain binary [34, 17, 6] codes, which includes some with trivial automorphism group.