E. Bannai, A. Munemasa, B. Venkov, “The nonexistence of certain tight spherical designs”, Algebra i Analiz, 16:4 (2004), 1–23; St. Petersburg Math. J., 16:4 (2005), 609

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Algebra i Analiz, 2004, Volume 16, Issue 4, Pages 1–23 (Mi aa616)

This article is cited in 52 scientific papers (total in 52 papers)

Research Papers

The nonexistence of certain tight spherical designs

E. Bannai^a, A. Munemasa^b, B. Venkov^c

^a Graduate school of Mathematics, Kyushu University, Fukuoka, Japan
^b Graduate School of Information Sciences, Tohoku University, Sendai, Japan
^c Steklov Institute of Mathematics at St. Petersburg, St. Petersburg, Russia

Full-text PDF (974 kB) Citations (52)

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Abstract: 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 = (2 m + 1)^{2} - 2

$n=(2m+1)^2-2$ , and existence is known only for

m = 1, 2

$m=1,2$ . In the paper, existence is ruled out under a certain arithmetic condition on the integer

m

$m$ , satisfied by infinitely many values of

m

$m$ , including

m = 4

$m=4$ . Also, nonexistence is shown for

m = 3

$m=3$ . Tight spherical 7-designs may exist in dimension

n = 3 d^{2} - 4

$n=3d^2-4$ , and existence is known only for

d = 2, 3

$d=2,3$ . In the paper, existence is ruled out under a certain arithmetic condition on

d

$d$ , satisfied by infinitely many values of

d

$d$ , including

d = 4

$d=4$ . Also, nonexistence is shown for

d = 5

$d=5$ . The fact that the above arithmetic conditions on

m

$m$ for 5-designs and on

d

$d$ for 7-designs are satisfied by infinitely many values of

m

$m$ ,

d

$d$ , respectively, is shown in the appendix written by Y.-F. S. Pétermann.

Received: 03.09.2003

English version:
St. Petersburg Mathematical Journal, 2005, Volume 16, Issue 4, Pages 609–625
DOI: https://doi.org/10.1090/S1061-0022-05-00868-X

Bibliographic databases:

Document Type: Article

Language: English

Citation: E. Bannai, A. Munemasa, B. Venkov, “The nonexistence of certain tight spherical designs”, Algebra i Analiz, 16:4 (2004), 1–23; St. Petersburg Math. J., 16:4 (2005), 609–625

Citation in format AMSBIB

\Bibitem{BanMunVen04}

\by E.~Bannai, A.~Munemasa, B.~Venkov

\paper The nonexistence of certain tight spherical designs

\jour Algebra i Analiz

\yr 2004

\vol 16

\issue 4

\pages 1--23

\mathnet{http://mi.mathnet.ru/aa616}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2090848}

\zmath{https://zbmath.org/?q=an:1072.05017}

\transl

\jour St. Petersburg Math. J.

\yr 2005

\vol 16

\issue 4

\pages 609--625

\crossref{https://doi.org/10.1090/S1061-0022-05-00868-X}

Linking options:

https://www.mathnet.ru/eng/aa616

https://www.mathnet.ru/eng/aa/v16/i4/p1

This publication is cited in the following 52 articles:

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Benjamin Nasmith, “Octonions and the two strictly projective tight 5-designs”, Algebraic Combinatorics, 5:3 (2022), 401
Iverson J.W., King E.J., Mixon D.G., “A Note on Tight Projective 2-Designs”, J. Comb Des., 29:12 (2021), 809–832
Bannai E., Bannai E., Xiang Z., Yu W.-H., Zhu Ya., “Classification of Spherical 2-Distance (4,2,1)-Designs By Solving Diophantine Equations”, Taiwan. J. Math., 25:1 (2021), 1–22
Boyvalenkov P., Stoyanova M., “Linear Programming Bounds For Covering Radius of Spherical Designs”, Results Math., 76:2 (2021), 95
Dostert M., De Laat D., Moustrou Ph., “Exact Semidefinite Programming Bounds For Packing Problems”, SIAM J. Optim., 31:2 (2021), 1433–1458
Gonzalez Merino B., Schymura M., “On the Reverse Isodiametric Problem and Dvoretzky-Rogers-Type Volume Bounds”, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 114:3 (2020), 136
Daniel Hughes, Shayne Waldron, “Spherical half-designs of high order”, Involve, 13:2 (2020), 193
King E.J., Tang X., “New Upper Bounds For Equiangular Lines By Pillar Decomposition”, SIAM Discret. Math., 33:4 (2019), 2479–2508
Fickus M., Jasper J., Mixon D.G., Peterson J.D., Watson C.E., “Equiangular Tight Frames With Centroidal Symmetry”, Appl. Comput. Harmon. Anal., 44:2 (2018), 476–496
Szollosi F., Ostergard P.R.J., “Enumeration of Seidel Matrices”, Eur. J. Comb., 69 (2018), 169–184
Glazyrin A., Yu W.-H., “Upper Bounds For S-Distance Sets and Equiangular Lines”, Adv. Math., 330 (2018), 810–833
Bannai E., Zhao D., Zhu L., Zhu Ya., Zhu Y., “Half of An Antipodal Spherical Design”, Arch. Math., 110:5 (2018), 459–466
Nozaki H., Suda Sh., “Complex Spherical Codes With Three Inner Products”, Discret. Comput. Geom., 60:2 (2018), 294–317
SeungHyun Shin, “Nonexistence of certain pseudogeometric graphs”, Discrete Mathematics, 341:4 (2018), 1125
Bannai E., Bannai E., Tanaka H., Zhu Ya., “Design Theory from the Viewpoint of Algebraic Combinatorics”, Graphs Comb., 33:1 (2017), 1–41
Foucart S., Skrzypek L., “On maximal relative projection constants”, J. Math. Anal. Appl., 447:1 (2017), 309–328
Yu W.-H., “New Bounds For Equiangular Lines and Spherical Two-Distance Sets”, SIAM Discret. Math., 31:2 (2017), 908–917
A. A. Makhnev, D. V. Paduchikh, M. M. Khamgokova, “Automorphisms of strongly regular graphs with parameters $(1305,440,115,165)$ ”, Proc. Steklov Inst. Math. (Suppl.), 304:1 (2019), S112–S122

Citing articles in Google Scholar: Russian citations, English citations
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