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<p>Euclidean <i>t</i>-designs are the two-step generalization of spherical <i>t</i>-designs, and coherent configurations are the extension of association schemes. It was proved that the spherical embedding of <i>Q</i>-polynomial association schemes can become spherical <i>t</i>-designs under certain conditions. In this paper, we extend the Sidelnikov inequality to Euclidean space and present an equivalent condition of Euclidean <i>t</i>-designs. As the main result, we propose a necessary and sufficient condition for the spherical embedding of a <i>Q</i>-polynomial coherent configuration (with two fibers) to be a Euclidean <i>t</i>-design (on two concentric spheres) for any positive integer <i>t</i>. In addition, we prove <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(t\le 10\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>≤</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation> if we further assume each fiber is a <i>P</i>-polynomial association scheme.</p>

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Euclidean designs obtained from Q-polynomial coherent configurations

  • 期刊论文

  • 发表日期：

    2026年6月30日

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  • Yuchen Jiang,
  • Yan Zhu,
  • Chengju Wu

摘要

Euclidean t-designs are the two-step generalization of spherical t-designs, and coherent configurations are the extension of association schemes. It was proved that the spherical embedding of Q-polynomial association schemes can become spherical t-designs under certain conditions. In this paper, we extend the Sidelnikov inequality to Euclidean space and present an equivalent condition of Euclidean t-designs. As the main result, we propose a necessary and sufficient condition for the spherical embedding of a Q-polynomial coherent configuration (with two fibers) to be a Euclidean t-design (on two concentric spheres) for any positive integer t. In addition, we prove \(t\le 10\) t ≤ 10 if we further assume each fiber is a P-polynomial association scheme.