Counterexample to the $l$-modular Belfiore-Sol\'e Conjecture

We show that the secrecy function conjecture that states that the maximum of the secrecy function of an $l$-modular lattice occurs at $1\mathrm{/}\sqrt{l}$ is false, by proving that the 4-modular lattice $C^(4) = $\mathbb{Z}$ \oplus \sqrt{2}$\mathbb{Z}$ \oplus 2$\mathbb{Z}$$ fails to satisfy this conjecture. We also indicate how the secrecy function must be modified in the $l$-modular case to have a more reasonable chance for it to have a maximum at $1\mathrm{/}\sqrt{l}$, and show that the conjecture, modified with this new secrecy function, is true for various odd 2-modular lattices.