从二元自对偶码构造的Construction-A格子的优超性与高斯质量极大性
Majorization and Gaussian-Mass Maximality for Construction-A Lattices from Binary Self-Dual Codes
AI总结：
本文证明，对于由二元自对偶码生成的所有幺模Construction-A格子，整数格在给定秩的整格子中最大化高斯质量，通过将theta级数不等式简化为码的优超性陈述，并利用凸序和Jensen不等式。
摘要：
Regev和Stephens-Davidowitz猜想整数格在给定秩的整格子中最大化高斯质量。我们证明了这一点，包括等式情况，对于所有由二元自对偶码生成的幺模Construction-A格子。证明将theta级数不等式简化为码的一个尖锐优超性陈述：如果$C$是一个二元自对偶$[2k,k]$码，那么$C$的半重量分布在凸序上被$\operatorname{Bin}(k,1/2)$控制，后者是$\mathbb{Z}^{2k}$的重复码模型的相应分布。实际上，将$C$置于系统形式$[I\mid A]$后，自对偶性给出$AA^T=I$在$\mathbb{F}_2$上，因此对于均匀随机消息$a$，两个重量$\wt(a)$和$\wt(aA)$具有相同的二项分布。所得码字的半重量是它们的平均值，Jensen不等式随后给出凸序控制。应用于构建theta级数的凸测试函数，这产生了高斯质量间隙的平方和公式；应用于铰链函数，它给出了简化间隙多项式的系数非负性。
英文摘要：
Regev and Stephens-Davidowitz conjectured that the integer lattice maximizes Gaussian mass among integral lattices of a given rank. We prove this, including the equality case, for all unimodular Construction-A lattices arising from binary self-dual codes. The proof reduces the theta-series inequality to a sharp majorization statement for codes: if $C$ is a binary self-dual $[2k,k]$ code, then the half-weight distribution of $C$ is dominated in convex order by $\operatorname{Bin}(k,1/2)$, which is the corresponding distribution for the repetition-code model of $\mathbb{Z}^{2k}$. Indeed, after putting $C$ in systematic form $[I\mid A]$, self-duality gives $AA^T=I$ over $\mathbb{F}_2$, so for a uniformly random message $a$ the two weights $\operatorname{wt}(a)$ and $\operatorname{wt}(aA)$ have the same binomial law. The half-weight of the resulting codeword is their average, and Jensen's inequality then gives convex-order domination. Applied to the convex test functions that build the theta series, this yields a sum-of-squares formula for the Gaussian-mass gap; applied to hinge functions, it gives coefficientwise nonnegativity of the reduced gap polynomial.
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