Arithmetic Nonexistence Results for Tight Spherical $5$-Designs

Source abstract

We prove two arithmetic nonexistence criteria for tight spherical $5$-designs in dimension $(2m+1)^2-2$ with $m$ even. Together, they recover the applicable criteria of Bannai-Munemasa-Venkov and Nebe-Venkov while covering parameters excluded by neither earlier result. Examples include $m=16,100$ under the first criterion and $m=96,168$ under the second. The proofs combine lattice moment identities, finite Gauss sums, and a determinant calculation modulo $3$.