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.
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机译：Belfiore-Sole猜想指出，对于R n 中的单模晶格Λ，将Z n 的θ系列的商除以Λ的θ系列而定。纯粹的虚值z = ty，y> 0，在y = 1处达到最大值。这个猜想与窃听者端的混乱紧密相关，涉及高斯信道的窃听代码。在本文中，我们证明了在单模晶格的保密函数上，有无数个晶格满足Belfiore-Solé猜想。我们进一步表明，构造A从长度最大为40的二进制，双偶，自对偶代码中获得的所有晶格都满足猜想。
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