Arithmetic Nonexistence Conditions for Tight Spherical $5$-Designs

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