On the secrecy gain of ℓ\ell-modular lattices

We show that for every $\ell>1$, there is a counterexample to the $\ell$-modular secrecy function conjecture by Oggier, Sol\'e and Belfiore. These counterexamples all satisfy the modified conjecture by Ernvall-Hyt\"onen 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 $\ell\in \{6,14,15\}$

Modular Lattices from a Variation of Construction A over Number Fields

We consider a variation of Construction A of lattices from linear codes based on two classes of number fields, totally real and CM Galois number fields. We propose a generic construction with explicit generator and Gram matrices, then focus on modular and unimodular lattices, obtained in the particular cases of totally real, respectively, imaginary, quadratic fields. Our motivation comes from coding theory, thus some relevant properties of modular lattices, such as minimal norm, theta series, kissing number and secrecy gain are analyzed. Interesting lattices are exhibited