What does this research mean for the field?

A linear-algebraic principle based on additive kernels with finitely many negative squares can be used to establish cardinality bounds for equilateral sets and equidistant codes, including exact bounds for squared Lee weight codes. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish a general linear-algebraic framework that bounds the size of equilateral sets and equidistant codes governed by additive kernels with finitely many negative squares.

September 5, 2026Open Access

Equidistant sets for additive kernels with finitely many negative squares

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LLuojunField Museum of Natural History

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Implication

Mathematical analysis demonstrates cardinality bounds on equidistant sets across additive kernels, establishing tight limits for Lee-metric codes and solving asymptotic Schütte problems.

Key Points

To establish a general linear-algebraic framework that bounds the size of equilateral sets and equidistant codes governed by additive kernels with finitely many negative squares.
Analyzed coordinatewise sums of anchored Schoenberg kernels possessing at most q negative squares on product spaces.
Decomposed Gram-type distance matrices into scaled centered matrices plus positive semidefinite terms of bounded rank (rank at most qn).
Formulated both exact rank inequalities and stable versions incorporating operator-norm errors and spectral-tail budgets, verified via analytic proofs and automated check scripts.
Established bounds for equidistant codes in Z_q^n under squared Lee weight of |C| <= (q/2)n + 1 for even q and |C| <= 2*ceil((q-1)/4)n + 1 for odd q without linearity assumptions, achieving tightness at q = 5 with |C| <= 2n + 1 for n = (5k - 1)/2.
Tightened the finite negative rank budget from 4kn to 2kn for p in (4k, 4k + 2) within the centered-Gram identity framework.
Derived the exact first-order asymptotic limit for the Schütte problem across 2 <= p <= 4 as lim_{N -> inf} N(R_p(N) - 1) = 2/p.

Cite This Study

Luojun (2026) studied this question.

synapsesocial.com/papers/6a9bd3726b95aff0620eabe7 https://doi.org/10.5281/zenodo.22272459

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Implication

Mathematical analysis demonstrates cardinality bounds on equidistant sets across additive kernels, establishing tight limits for Lee-metric codes and solving asymptotic Schütte problems.

Equidistant sets for additive kernels with finitely many negative squares

Discussion

Member takes

Implication

Key Points

Cite This Study

Equidistant sets for additive kernels with finitely many negative squares

Discussion

Member takes

Implication

Key Points

Cite This Study

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