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arxiv: 2606.03482 · v2 · pith:3OMOP7NOnew · submitted 2026-06-02 · 🧮 math.NT · cs.IT· math.CO· math.IT· math.MG

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

Scott Duke Kominers This is my paper

Pith reviewed 2026-06-28 08:34 UTC · model grok-4.3

classification 🧮 math.NT cs.ITmath.COmath.ITmath.MG
keywords Construction-A latticesbinary self-dual codesGaussian massmajorizationconvex ordertheta seriesJensen inequalityunimodular lattices
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The pith

The integer lattice maximizes Gaussian mass among all unimodular Construction-A lattices from binary self-dual codes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Regev and Stephens-Davidowitz conjectured that the integer lattice maximizes Gaussian mass among integral lattices of fixed rank. This paper proves the conjecture holds, including equality cases, for every unimodular lattice obtained by Construction A from a binary self-dual code. The proof reduces the theta-series comparison to a majorization claim: the half-weight distribution of any such code is dominated in convex order by the binomial distribution that arises from the repetition code. Self-duality in systematic form forces the two component weights of a random codeword to share the same binomial law, so their average obeys Jensen's inequality for the convex functions that appear in the theta series. This produces both a sum-of-squares formula for the mass gap and coefficientwise nonnegativity for the reduced gap polynomial.

Core claim

For every binary self-dual [2k,k] code C, the half-weight distribution of C is dominated in convex order by Bin(k,1/2). Consequently every unimodular Construction-A lattice built from such a code has Gaussian mass at most that of the integer lattice Z^{2k}, with equality precisely when C is equivalent to the repetition code. Placing C in systematic form [I|A] with AA^T = I over F_2 makes the two weight components identically distributed as Bin(k,1/2) for uniform random messages; the half-weight is their average, and Jensen's inequality applied to the convex test functions of the theta series therefore yields the required domination.

What carries the argument

Convex-order majorization of the half-weight distribution by Bin(k,1/2), obtained because self-duality equates the marginal weight distributions and Jensen's inequality then applies to the convex functions in the theta series.

If this is right

  • The Gaussian-mass gap admits a sum-of-squares expression.
  • The reduced gap polynomial has all coefficients nonnegative.
  • Equality holds in the mass inequality precisely when the code is equivalent to the repetition code.
  • The result applies to every dimension admitting a binary self-dual code.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same majorization argument may extend to weight distributions arising from other code families or lattice constructions.
  • Gaussian-mass extremality questions can be reduced to classical coding inequalities on weight distributions.
  • The technique supplies an explicit combinatorial certificate for the mass gap that might be checked in small dimensions.

Load-bearing premise

The Gaussian-mass comparison for these lattices reduces exactly to convex-order majorization of the half-weight distribution via Jensen's inequality on the convex functions that build the theta series.

What would settle it

A binary self-dual code whose half-weight distribution fails to be convex-order dominated by Bin(k,1/2), or an explicit Construction-A lattice whose Gaussian mass exceeds that of the integer lattice of the same rank.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that every unimodular Construction-A lattice arising from a binary self-dual [2k,k] code has Gaussian mass at most that of the integer lattice Z^{2k}, with equality characterized. The argument places the code in systematic form [I|A] with AA^T = I over F_2, observes that the two coordinate weights are marginally Bin(k,1/2), and invokes Jensen's inequality on their average (the half-weight) to obtain convex-order domination by the binomial distribution; this domination is then applied to the convex functions appearing in the theta-series expansion, yielding an explicit sum-of-squares formula for the mass gap together with coefficientwise non-negativity of the reduced gap polynomial via hinge functions.

Significance. The result establishes the Regev-Stephens-Davidowitz conjecture for an infinite family of lattices and supplies a parameter-free, explicit positive representation of the gap. The derivation relies only on the self-duality relation and standard convexity, with no fitted parameters or ad-hoc constructions.

minor comments (2)
  1. [Introduction] The definition of the half-weight random variable (average of wt(a) and wt(aA)) is used throughout but would benefit from an explicit displayed equation in the introduction or §2.
  2. [Theorem 1.1] The equality case in the majorization statement is asserted but the precise condition on the code (when the two weight random variables are a.s. equal) is not isolated as a separate corollary; adding this would clarify the characterization.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and for the positive recommendation to accept the manuscript. The report accurately captures the main argument and its consequences.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation chain begins from the self-duality condition AA^T = I over F_2 for systematic form [I|A] of a binary self-dual code, which directly implies that wt(a) and wt(aA) are each marginally Bin(k,1/2) for uniform random a. Jensen's inequality is then applied to the convex functions appearing in the theta-series expansion of the Construction-A lattice, yielding convex-order majorization of the half-weight distribution by Bin(k,1/2) and an explicit sum-of-squares gap formula. All steps are parameter-free, rest on standard properties of self-dual codes and convex analysis, and contain no fitted inputs, self-citations, or renamings that reduce the claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on standard properties of self-dual codes and convex analysis; no free parameters or new entities are introduced.

axioms (2)
  • standard math Jensen's inequality for convex functions
    Applied to obtain convex-order domination of the half-weight distribution from the equal binomial laws of wt(a) and wt(aA).
  • domain assumption Self-dual codes in systematic form satisfy AA^T = I over F_2
    This ensures the two weight random variables have identical distributions.

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Reference graph

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