On Tight Euclidean 6-Designs: An Experimental Result

Djoko Suprianto

doi:10.5614/itbj.sci.2011.43.1.3

Published: 2011-01-01 | DOI: 10.5614/itbj.sci.2011.43.1.3 | Vol 43, Issue 1 (2011)

Authors

Djoko Suprianto

Combinatorial Mathematics Group Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung 40132, INDONESIA.

Abstract

A finite set n X âŠ† â„n with a weight function w: X â†’ â„ > 0 is called Euclidean t-design in â„ > 0 (supported by p concentric spheres) if the following condition holds: ï€¨ ï€© ï€¨ ï€© ï€¨ ï€© ï€¨ ï€© ï€¨ ï€© 1 i p i i i S x X w X f d w f S ï³ ï€½ ïƒŽ ïƒ¥ ïƒ² x x ï€½ïƒ¥ x x for any polynomial f(x) âˆˆ Polâ„ > 0 of degree at most t. Here Si â„n is a sphere of radius ri â‰¥ 0, Xi=X âˆ© S, and Ïƒi(x) is an O(n) -invariant measure on Si such that |Si|=rin-1|Sn-1>|, with |Si| is the surface area of Si and |Sn-1|is a surface area of the unit sphere in â„n. Recently, Bajnok [1] constructed tight Euclidean t-designs in the plane (n=2) for arbitrary t and p. In this paper we show that for case t=6 and p=2, tight Euclidean 6-designs constructed by Bajnok is the unique configuration in â„œn, for 2 â‰¤ n â‰¤ 8.

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References

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Seidel, J.J., Designs and Approximation, Contemp. Math., 111, pp. 179- 186, 1990.

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Bannai, E. & Bannai, Et., On Euclidean Tight 4-Designs, J. Math. Soc. Japan, 58(3), pp. 775-804, 2006.

Bannai, E., Bannai, Et. & Suprijanto, Dj., On The Strong Non-Rigidity of Certain Tight Euclidean Designs, European J. Combin., 28(6), pp. 1662-1680, 2007.

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Bannai, Et., New examples of Euclidean Tight 4-Designs, European J. Combin., 30(3), 655-667, 2009.

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Bannai, E. & Damerell, R.M., Tight Spherical Designs I, J. Math. Soc. Japan, 31(1), pp. 199-207, 1979.

Boyvalenkov, P. & Nikova, S., Improvements of The Lower Bounds on The Size of Some Spherical Designs, Math. Balkanica, 12(1-2), pp.151-160, 1998.

Bannai, E. & Bannai, Et., Tight Gaussian 4-Designs, J. Alg. Combin., 22(1), pp. 39-63, 2005.

Bajnok, B., Orbits of The Hyperoctahedral Group as Euclidean Designs, J. Alg. Combin., 25(4), pp. 375-397, 2007.

Bannai, E., Bannai, Et., Hirao, M. & Sawa, M., Cubature Formulas in Numerical Analysis and Tight Euclidean Designs, European J. Combin., 31(2), pp. 423-441, 2010.

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[ref-1] Bajnok, B., On Euclidean t-designs, Adv. Geom., 6(3), 423-438, 2006.

[ref-2] Seidel, J.J., Designs and Approximation, Contemp. Math., 111, pp. 179- 186, 1990.

[ref-3] Delsarte, P., Goethals, J.-M. & Seidel, J.J., Spherical Codes and Designs, Geom. Dedicata, 6(3), pp. 363-388, 1977.

[ref-4] Neumaier, A. & Seidel, J.J., Discrete Measures for Spherical Designs, Eutactic Stars and Lattices, Nederl. Akad. Wetensch. Proc. Ser. A 91= Indag. Math., 50(3), pp. 321-334, 1988.

[ref-5] Delsarte, P. & Seidel, J.J., Fisher Type Inequalities for Euclidean t- Designs, Linear Algebra Appl., 114-115, 213-230, 1989.

[ref-6] Bannai, E. & Bannai, Et., On Euclidean Tight 4-Designs, J. Math. Soc. Japan, 58(3), pp. 775-804, 2006.

[ref-7] Bannai, E., Bannai, Et. & Suprijanto, Dj., On The Strong Non-Rigidity of Certain Tight Euclidean Designs, European J. Combin., 28(6), pp. 1662-1680, 2007.

[ref-8] Bannai, E. & Ito, T., Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, CA, 1984.

[ref-9] Delarte, P., An Algebraic Approach to The Association Schemes of Coding Theory, Philips Res. Repts Suppl., 10, 1973.

[ref-10] Bannai, E. & Bannai, Et., A Note on The Spherical Embedding of Strongly Regular Graphs, European J. Combin., 26(8), pp. 1177-1179, 2005.

[ref-11] Bannai, E. & Bannai, Et., On Primitive Symmetric Association Schemes with 1 m ï€1/2 3, Contribution Disc. Math., 1(1), pp. 68-79, 2006.

[ref-12] Bannai, E. & Bannai, Et., Algebraic Combinatorics on Spheres (in Japanese) Springer Tokyo 1999.

[ref-13] Ericson, T. & Zinoviev, V., Codes on Euclidean Spaces, North-Holland, 2001.

[ref-14] Bannai, Et., New examples of Euclidean Tight 4-Designs, European J. Combin., 30(3), 655-667, 2009.

[ref-15] Bannai, E. & Bannai, Et., On Antipodal Spherical t-Designs of Degree s with t "žd 2s Æ

[ref-16] Bannai, Et., On Antipodal Euclidean Tight (2e Æ

[ref-17] Erdelyi, A., et al., Higher Transcendental Functions, Vol. II, (Bateman Manuscript Project), MacGraw-Hill, 1953.

[ref-18] van Dam, E.R., Three-Class Association Schemes, J. Alg. Combin., 10(1), pp. 69-107, 1999.

[ref-19] Bannai, E. & Damerell, R.M., Tight Spherical Designs I, J. Math. Soc. Japan, 31(1), pp. 199-207, 1979.

[ref-20] Boyvalenkov, P. & Nikova, S., Improvements of The Lower Bounds on The Size of Some Spherical Designs, Math. Balkanica, 12(1-2), pp.151-160, 1998.

[ref-21] Bannai, E. & Bannai, Et., Tight Gaussian 4-Designs, J. Alg. Combin., 22(1), pp. 39-63, 2005.

[ref-22] Bajnok, B., Orbits of The Hyperoctahedral Group as Euclidean Designs, J. Alg. Combin., 25(4), pp. 375-397, 2007.

[ref-23] Bannai, E., Bannai, Et., Hirao, M. & Sawa, M., Cubature Formulas in Numerical Analysis and Tight Euclidean Designs, European J. Combin., 31(2), pp. 423-441, 2010.

[ref-24] Bannai, E., Bannai, Et. & Shigezumi, J., A New Example of Euclidean Tight 6-design, preprint, 2010.