Arithmetic Nonexistence Conditions for Tight Spherical $5$-Designs

Abstract

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 ${\nu }_2(m(m+1))\le 5$, where ${\nu }_p(a)$ denotes the exponent of the prime $p$ in the positive integer $a$. We show that such a design does not exist if either (i) $m\equiv 1(\mathrm{mod}3)$ and ${\nu }_p(m(m+1))\le 1$ for every prime $p\ge 7$ with $p\equiv 3(\mathrm{mod}4)$, or (ii) $m\equiv 0(\mathrm{mod}3)$, ${\nu }_3(m(m+1))=1$, and ${\nu }_p(m(m+1))\le 1$ for every prime $p\ge 5$ with $p\not\equiv 1(\mathrm{mod}12)$. 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.