发表机构
East China Normal University（华东师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对偶数$m$的维度$(2m+1)^2-2$中的紧球面$5$-设计，提出了两个算术不存在性准则，结合格矩恒等式、有限高斯和及模$3$行列式计算，覆盖了先前准则未涵盖的参数。
AI 中文摘要
我们证明了在维度$(2m+1)^2-2$（其中$m$为偶数）中紧球面$5$-设计的两个算术不存在性准则。这两个准则共同恢复了Bannai-Munemasa-Venkov和Nebe-Venkov的适用准则，同时覆盖了先前两个结果均未涵盖的参数。例子包括第一个准则下的$m=16,100$和第二个准则下的$m=96,168$。证明结合了格矩恒等式、有限高斯和以及模$3$的行列式计算。
英文摘要
We prove two arithmetic nonexistence conditions for tight spherical $5$-designs in dimension $(2m+1)^2-2$. Let $m$ be a positive even integer satisfying $ν_2(m(m+1))\leq5$, where $ν_p(a)$ denotes the exponent of the prime $p$ in the positive integer $a$. We show that such a design does not exist if either (i) $m\equiv1\pmod3$ and $ν_p(m(m+1))\leq1$ for every prime $p\geq7$ with $p\equiv3\pmod4$, or (ii) $m\equiv0\pmod3$, $ν_3(m(m+1))=1$, and $ν_p(m(m+1))\leq1$ for every prime $p\geq5$ with $p\not\equiv1\pmod{12}$. Together, these conditions recover the corresponding nonexistence results of Bannai, Munemasa, and Venkov and of Nebe and Venkov, and exclude parameters not covered by either earlier result, including $m=16,40,88,96,100$. The proofs use lattice theory to constrain the discriminant groups of the associated lattices, and derive contradictions through Gauss sums and determinant congruences.
