The nonexistence of certain tight spherical designs

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 = 3d² − 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.