Majorization and Gaussian-Mass Maximality for Construction-A Lattices from Binary Self-Dual Codes

Abstract

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.