The nonexistence of certain tight spherical designs

Abstract

In this paper, the nonexistence of tight spherical designs is shown in some cases left open to date. Tight spherical 5-designs may exist in dimension $n=(2m+1)^{2}-2$, and the existence is known only for $m=1,2$. In the paper, the existence is ruled out under a certain arithmetic condition on the integer $m$, satisfied by infinitely many values of $m$, including $m=4$. Also, nonexistence is shown for $m=3$. Tight spherical 7-designs may exist in dimension $n=3d^{2}-4$, and the existence is known only for $d=2,3$. In the paper, the existence is ruled out under a certain arithmetic condition on $d$, satisfied by infinitely many values of $d$, including $d=4$. Also, nonexistence is shown for $d=5$. The fact that the arithmetic conditions on $m$ for $5$-designs and on $d$ for $7$-designs are satisfied by infinitely many values of $m$ and $d$, respectively, is shown in the Appendix written by Y.-F. S. Pétermann.