Title: 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 $\ell\in \{6,14,15\}$.