TY - JOUR

T1 - The nonexistence of certain tight spherical designs

AU - Bannai, E.

AU - Munemasa, A.

AU - Venkov, B.

PY - 2005

Y1 - 2005

N2 - 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.

AB - 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.

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U2 - 10.1090/S1061-0022-05-00868-X

DO - 10.1090/S1061-0022-05-00868-X

M3 - Article

AN - SCOPUS:85009737086

SN - 1061-0022

VL - 16

SP - 609

EP - 625

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

IS - 4

ER -