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The Sharp Exponential Constant for 6-Regular Circulant Matrices

A mix of human-written and AI-generated textHuman understanding: all partsmath.CO — Combinatorics

Contributed by Yair Lavi ↗

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

We prove that, for every integer n≥6n\ge6 and six-element set S⊆ZnS\subseteq\mathbb Z_n, the binary circulant Cn(S)=(1{j−i∈S})i,j∈ZnC_n(S)=(1_{\{j-i\in S\}})_{i,j\in\mathbb Z_n} satisfies per⁡Cn(S)≥((3125/1296)e7/250000)n\operatorname{per}C_n(S)\ge((3125/1296)e^{7/250000})^n. For every integer d≥2d\ge2, we identify the optimal uniform base as inf⁡n≥dinf⁡∣S∣=dper⁡Cn(S)1/n=lim⁡q→∞per⁡Cqd−1({0,1,q,…,qd−2})1/qd−1\inf_{n\ge d}\inf_{|S|=d}\operatorname{per}C_n(S)^{1/n}=\lim_{q\to\infty}\operatorname{per}C_{q^{d-1}}(\{0,1,q,\ldots,q^{d-2}\})^{1/q^{d-1}}, where S⊆ZnS\subseteq\mathbb Z_n and qq runs through integers at least 22. In degree six this base is at most 1638128736631981/35163812873663198^{1/35}, while the lower bound strictly exceeds the sharp unrestricted base 3125/12963125/1296. This resolves the surviving circulant lower-bound question in Minc's Problem 10; its proposed full upper bound fails even for circulants.

Author comments: The supplement cited in the text (supplement-v7.tar.gz) is in the Zenodo record https://doi.org/10.5281/zenodo.23044484.

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References

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Version history

  1. v1 · 15 pagesSubmitted by Yair LaviThe supplement cited in the text (supplement-v7.tar.gz) is in the Zenodo record https://doi.org/10.5281/zenodo.23044484.CurrentNov 13, 2026