On the secrecy gain of $\ell$-modular lattices

Abstract

We show that for every $\ell >1$, there is a counterexample to the $\ell$-modular secrecy function conjecture by Oggier, Solé and Belfiore. These counterexamples all satisfy the modified conjecture by Ernvall-Hytönen and Sethuraman. Furthermore, we provide a method to prove or disprove the modified conjecture for any given $\ell$-modular lattice rationally equivalent to a suitable amount of copies of $\mathbb{Z}\oplus \sqrt{\ell }\mathbb{Z}$ with $\ell \in \{3,5,7,11,23\}$. We also provide a variant of the method for strongly $\ell$-modular lattices when $\ell \in \{6,14,15\}$.

References in corpus (1)

• Unimodular Lattices for the Gaussian Wiretap Channel

Cited by in corpus (1)

• On a conjecture of Faulhuber and Steinerberger on the logarithmic derivative of ${\vartheta }_4$