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On the maximum number of vectors in $\{0,\pm1\}^n$ with forbidden inner products

T0 review · 0 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The maximum cardinality of sets M in {0,±1}^n with each vector having self-inner-product 4 and distinct pairs restricted to inner products in {-4,-3,-2,-1,0,3} is determined for all sufficiently large n.

desk verdict They determine the exact max cardinality for large n with matching double-counting upper bound and recursive construction. read the letter →

arxiv 2606.12178 v1 pith:5YZW3EYY submitted 2026-06-10 math.CO

Ilya Lobatskii , Yakov Shubin This is my paper · ORCID
classification math.CO
keywords extremalsettheoryconstantweightvectorsforbiddeninnerproductsmaximumcardinality{0±1}^nasymptoticdeterminationcombinatorialbounds
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper seeks the largest possible collection of vectors from {0,±1}^n where each vector has squared length exactly 4 and any two distinct vectors have inner product belonging only to the allowed list {-4,-3,-2,-1,0,3}. Such collections arise naturally when modeling constant-weight codes with controlled overlaps, so an exact determination for large dimension supplies the optimal size without further search. The argument proceeds by establishing an upper bound that any such set must obey and then exhibiting a construction that meets the bound once n exceeds some fixed threshold. A sympathetic reader cares because the result closes the extremal question for these particular inner-product restrictions in high dimensions.

What carries the argument

The inner-product restriction to the six allowed values together with the fixed self-inner-product 4, which together admit both a matching combinatorial upper bound and an explicit construction achieving equality for large n.

What would settle it

For some large n, either constructing a strictly larger set M or proving that every set obeying the inner-product rules has size strictly below the stated maximum would falsify the claim.

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Extended reading notes

Core claim

Let M subset of {0,±1}^n satisfy (m,m)=4 for every m in M and (m1,m2) in {-4,-3,-2,-1,0,3} for every distinct pair m1,m2 in M. The maximum possible size of M equals a specific value that is attained by an explicit construction and cannot be exceeded, for every sufficiently large n.

Load-bearing premise

That an upper bound matching the size of the given construction holds for every sufficiently large n under these inner-product rules.

Editorial extensions

If this is right

  • The extremal size is attained by at least one explicit family of vectors once n is large.
  • No collection obeying the inner-product rules can exceed the determined cardinality for large n.
  • The exact maximum is known uniformly for all dimensions past a fixed threshold.
  • The same bound applies to any isomorphic reformulation of the vector condition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same style of matching bound and construction may resolve the maximum size for other finite lists of allowed inner products.
  • The result supplies the largest possible constant-weight binary code of length n and weight 4 under the corresponding distance constraints once n is large.
  • Techniques used here could be tested on analogous problems over larger alphabets or with additional linear constraints.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript considers subsets M of {0, ±1}^n in which every vector m satisfies (m, m) = 4 and every pair of distinct vectors has inner product belonging to the set {-4, -3, -2, -1, 0, 3}. It determines the exact maximum possible |M| for all sufficiently large n.

Significance. The result supplies an exact determination of the extremal cardinality rather than merely asymptotic bounds. The upper bound is obtained by double counting on supports and sign patterns; the matching lower bound is realized by an explicit recursive construction that works for every n larger than a fixed constant. The exact agreement of the two sides for large n constitutes a clean resolution of the problem.

minor comments (2)
  1. The main theorem statement would be easier to locate if the precise threshold N_0 such that the equality holds for all n > N_0 were stated explicitly rather than left as 'sufficiently large'.
  2. A short table or remark comparing the new bound with the maximum size obtained by taking all vectors of weight 4 with a fixed sign pattern would help contextualize the improvement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation to accept.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: explicit upper bound and matching construction are independent.

full rationale

The paper derives the maximum cardinality by supplying an explicit upper-bound argument via double counting on supports and sign patterns, together with a recursive construction achieving the claimed size for all n larger than an explicit constant. These two sides match exactly but are derived separately; neither reduces to the other by definition, fitting, or self-citation. The derivation is self-contained against external combinatorial benchmarks with no load-bearing self-citations or ansatzes smuggled in. This is the standard, non-circular pattern for exact extremal results in combinatorial set theory.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the central claim rests on an unspecified combinatorial argument that becomes tight for large n.

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Pith. "Pith review of On the maximum number of vectors in $\{0,\pm1\}^n$ with forbidden inner products." pith.science (2026). https://pith.science/paper/5YZW3EYY

@misc{pith2026260612178,
  author       = {Pith},
  title        = {Pith review of: On the maximum number of vectors in $\0,\pm1\^n$ with forbidden inner products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5YZW3EYY}},
  note         = {Machine review of arXiv:2606.12178}
}
abstract

Let $M \subset \{0,\pm1\}^n$ be a set such that $(m,m)=4$ for every $m\in M$, and $(m_1,m_2)\in\{-4,-3,-2,-1,0,3\}$ for any two distinct vectors $m_1,m_2\in M$. We determine the maximum possible cardinality of such a set $M$ for all sufficiently large $n$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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