Majorization and Gaussian-Mass Maximality for Construction-A Lattices from Binary Self-Dual Codes
2026-06-02 • Information Theory
Information Theory
AI summaryⓘ
The authors prove a mathematical guess by Regev and Stephens-Davidowitz about a lattice called the integer lattice having the highest Gaussian mass among certain lattices built from binary self-dual codes. They focus on Construction-A lattices and show this by comparing weight distributions in the codes to a known binomial distribution. Using properties from coding theory and inequalities, they demonstrate this comparison leads to the lattice maximizing the Gaussian mass, even identifying when equality holds. Their approach connects code weights to the lattice’s Gaussian mass through convex order and theta series techniques.
integer latticeGaussian massConstruction-A latticesbinary self-dual codestheta seriesconvex orderbinomial distributionJensen's inequalityweight distributionunimodular lattice
Authors
Scott Duke Kominers
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 $\wt(a)$ and $\wt(aA)$ 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.