Wiretap codes: Families of lattices satisfying the Belfiore-Solé secrecy function conjecture

The Belfiore-Sole conjecture states that for a unimodular lattice Λ in Rⁿ, the quotient of the theta series of Zⁿ by the theta series of Λ, when restricted to the purely imaginary values z = ty, y > 0, attains its maximum at y = 1. This conjecture is vitally connected to the confusion at the eavesdropper´s end in wiretap codes for Gaussian channels. In this paper we show that infinitely many lattices satisfy the Belfiore-Solé conjecture on the secrecy function of unimodular lattices. We further show that all lattices obtained by Construction A from binary, doubly even, self-dual codes of lengths up to 40 satisfy the conjecture.