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

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.

💡 Research Summary

The paper investigates the relationship between the kissing number of a lattice and its secrecy gain in the context of Gaussian wiretap coding. The authors focus first on even unimodular lattices of dimension (n = 24m + 8k) whose shortest non‑zero vector has squared length at least (2m). By expressing the secrecy gain (\chi(\Lambda) = \Theta_{\mathbb{Z}^n}(iy)/\Theta_{\Lambda}(iy)) in terms of the theta series of the lattice, they show that the coefficient (A_{2m}) – the number of vectors of length (2m) (the kissing number) – appears as the leading non‑trivial term in the denominator. Using modular transformation properties, combinatorial identities, and the Cauchy–Schwarz inequality, they prove that (\chi(\Lambda)) is a monotone decreasing function of (A_{2m}); consequently, reducing the kissing number strictly increases the secrecy gain.

The second part extends the argument to general unimodular lattices, which may contain both even and odd components. Although higher‑order coefficients (A_{2m+2}, A_{2m+4},\dots) also influence the theta series, the authors demonstrate that the dominant effect still comes from the shortest‑length vectors. By bounding the contributions of the higher‑order terms, they obtain an analogous monotonicity result: for any unimodular lattice with the same minimal norm, a smaller number of minimal vectors yields a larger secrecy gain.

Assuming the Belfiore–Solé conjecture that the secrecy gain is maximized at a specific value of the auxiliary parameter (y), the paper derives an explicit linear relation for the difference of the inverses of secrecy gains of two lattices that share the same dimension and minimal norm: \