## Taiwanese Journal of Mathematics

### Classification of Spherical $2$-distance $\{4,2,1\}$-designs by Solving Diophantine Equations

#### Abstract

In algebraic combinatorics, the first step of the classification of interesting objects is usually to find all their feasible parameters. The feasible parameters are often integral solutions of some complicated Diophantine equations, which cannot be solved by known methods. In this paper, we develop a method to solve such Diophantine equations in $3$ variables. We demonstrate it by giving a classification of finite subsets that are spherical $2$-distance sets and spherical $\{4,2,1\}$-designs at the same time.

#### Article information

Source
Taiwanese J. Math., Advance publication (2020), 22 pages.

Dates
First available in Project Euclid: 25 June 2020

https://projecteuclid.org/euclid.twjm/1593050477

Digital Object Identifier
doi:10.11650/tjm/200601

#### Citation

Bannai, Eiichi; Bannai, Etsuko; Xiang, Ziqing; Yu, Wei-Hsuan; Zhu, Yan. Classification of Spherical $2$-distance $\{4,2,1\}$-designs by Solving Diophantine Equations. Taiwanese J. Math., advance publication, 25 June 2020. doi:10.11650/tjm/200601. https://projecteuclid.org/euclid.twjm/1593050477

#### References

• E. Bannai and E. Bannai, A survey on spherical designs and algebraic combinatorics on spheres, European J. Combin. 30 (2009), no. 6, 1392–1425.
• E. Bannai and R. M. Damerell, Tight spherical designs I, J. Math. Soc. Japan 31 (1979), no. 1, 199–207.
• ––––, Tight spherical designs II, J. London Math. Soc. (2) 21 (1980), no. 1, 13–30.
• E. Bannai and T. Ito, Algebraic Combinatorics I: Association schemes, The Benjamin/Cummings, Menlo Park, CA, 1984.
• E. Bannai, A. Munemasa and B. Venkov, The nonexistence of certain tight spherical designs, St. Petersburg Math. J. 16 (2005), no. 4, 609–625.
• E. Bannai, T. Okuda and M. Tagami, Spherical designs of harmonic index $t$, J. Approx. Theory 195 (2015), 1–18.
• E. Bannai, D. Zhao, L. Zhu, Y. Zhu and Y. Zhu, Half of an antipodal spherical design, Arch. Math. (Basel) 110 (2018), no. 5, 459–466.
• A. Barg, A. Glazyrin, K. A. Okoudjou and W.-H. Yu, Finite two-distance tight frames, Linear Algebra Appl. 475 (2015), 163–175.
• P. G. Boyvalenkov, P. D. Dragnev, D. P. Hardin, E. B. Saff and M. M. Stoyanova, Universal upper and lower bounds on energy of spherical designs, Dolomites Res. Notes Approx. 8 (2015), Special Issue, 51–65.
• A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-regular Graphs, Ergebnisse der Mathematik und ihrer Grenzgebiet (3) 18, Springer-Verlag, Berlin, 1989.
• P. J. Cameron, J.-M. Goethals and J. J. Seidel, The Krein condition, spherical designs, Norton algebras and permutation groups, Nederl. Akad. Wetensch. Indag. Math. 40 (1978), no. 2, 196–206.
• H. Cohn and A. Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007), no. 1, 99–148.
• P. Delsarte, J. M. Goethals and J. J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977), no. 3, 363–388.
• G. Nebe and B. Venkov, On tight spherical designs, St. Petersburg Math. J. 24 (2013), no. 3, 485–491.
• A. Neumaier, New inequalities for the parameters of an association scheme, in: Combinatorics and Graph Theory, 365–367, Lecture Notes in Math. 885, Springer, Berlin, 1981.
• H. Nozaki and S. Suda, Bounds on $s$-distance sets with strength $t$, SIAM J. Discrete Math. 25 (2011), no. 4, 1699–1713.
• T. Okuda and W.-H. Yu, A new relative bound for equiangular lines and nonexistence of tight spherical designs of harmonic index $4$, European J. Combin. 53 (2016), 96–103.
• Z. Xiang, Nonexistence of nontrivial tight $8$-designs, J. Algebraic Combin. 47 (2018), no. 2, 301–318.
• ––––, Mathematica code to solve a certain degree $10$ Diophantine equation in $3$ variables under some conditions, Retrieved from the website: http://ziqing.org/.
• Y. Zhu, E. Bannai, E. Bannai, K.-T. Kim and W.-H. Yu, On spherical designs of some harmonic indices, Electron. J. Combin. 24 (2017), no. 2, 28 pp.