A short note on the kissing number of the lattice in Gaussian wiretap coding

TL;DR

This paper investigates how the secrecy gain of certain unimodular lattices in Gaussian wiretap coding relates to the number of shortest vectors, providing theoretical results and examples of lattices with identical parameters but different secrecy gains.

Contribution

It establishes a relationship between shortest vector count and secrecy gain in unimodular lattices and demonstrates that similar lattice parameters can yield different secrecy gains.

Findings

Secrecy gain increases as the number of shortest vectors decreases.

A formula relating secrecy gain differences to vector counts under a conjecture.

Existence of lattices with identical parameters but different secrecy gains.

Abstract

We show that on an $n=24m+8k$-dimensional even unimodular lattice, if the shortest vector length is $\geq 2m$, then as the number of vectors of length $2m$ decreases, the secrecy gain increases. We will also prove a similar result on general unimodular lattices. Furthermore, assuming the conjecture by Belfiore and Sol\'e, we will calculate the difference between inverses of secrecy gains as the number of vectors varies. Finally, we will show by an example that there exist two lattices in the same dimension with the same shortest vector length and the same kissing number, but different secrecy gains.