The nonexistence of certain tight spherical designs

In this paper, the nonexistence of tight spherical designs is shown in some cases left open to the date. Tight spherical 5-designs may exist in dimension $n=(2m+1)^2-2$, and existence is known only for $m=1,2$. In the paper, 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 existence is known only for $d=2,3$. In the paper, 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 above arithmetic conditions on $m$ for 5-designs and on $d$ for 7-designs are satisfied by infinitely many values of $m$, $d$, respectively, is shown in the appendix written by Y.-F. S. Pétermann.