Abstract
We investigate the structure of spherical τ-designs by applying polynomial techniques for investigation of some inner products of such designs. Our approach can be used for large variety of parameters (dimension, cardinality, strength). We obtain new upper bounds for the largest inner product, lower bounds for the smallest inner product and some other bounds. Applications are shown for proving nonexistence results either in small dimensions and in certain asymptotic process. In particular, we complete the classification of the cardinalities for which 3-designs on \({\mathbb{S}^{n-1}}\) exist for n = 8, 13, 14 and 18. We also obtain new asymptotic lower bound on the minimum possible odd cardinality of 3-designs.
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Abramowitz M., Stegun I.: Handbook of Mathematical Functions. Dover, New York (1965)
Bajnok B.: Constructions of spherical 3-designs. Graph. Combinator. 14, 97–107 (1998)
Bajnok B.: Spherical designs and generalized sum-free sets in abelian groups. Des. Codes Cryptogr. 21, 11–18 (2000)
Boumova S., Boyvalenkov P., Kulina H., Stoyanova M.: New Nonexistence Results for Spherical 5-Designs. Scientific Research, A Journal of South-Western University, Blagoevgrad, Bulgaria, 14 pp (2007).
Boyvalenkov P., Bumova S., Danev D.: Necessary conditions for existence of some designs in polynomial metric spaces. Europ. J. Combin. 20, 213–225 (1999)
Boumova S., Boyvalenkov P., Danev D.: New nonexistence results for spherical designs. In: Bojanov, B. (eds) Constructive Theory of Functions., pp. 225–232. Darba, Sofia (2003)
Boumova S., Boyvalenkov P., Danev D.: Recent results on the existence of spherical 3-designs. Presented at AMS Meeting, Baltimore, MD, unpublished (2003).
Boyvalenkov P., Danev D., Nikova S.: Nonexistence of certain spherical designs of odd strengths and cardinalities. Discr. Comp. Geom. 21, 143–156 (1999)
Conway J.H., Sloane N.J.A.: Sphere Packings, Lattices and Groups. Springer-Verlag, New York (1988)
Delsarte P., Goethals J.-M., Seidel J.J.: Spherical codes and designs. Geom. Dedicata 6, 363–388 (1977)
Hardin R.H., Sloane N.J.A.: McLaren’s improved snub cube and other new spherical designs in three dimensions. Discr. Comp. Geom. 15, 429–441 (1996)
Levenshtein V.I.: Universal bounds for codes and designs. In: Pless V., Huffman W.C. (eds.) Handbook of Coding Theory, Chapt. 6, pp. 499–648. Elsevier Science B.V. (1998).
Reznick B.: Some constructions of spherical 5-designs. Lin. Alg. Appl. 226/228, 163–196 (1995)
http://www.fmi.uni-sofia.bg/algebra/publications/stoyanova/table.html and http://www.moi.math.bas.bg/~silvi/table.html.
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Communicated by C.J. Colbourn.
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Boumova, S., Boyvalenkov, P., Kulina, H. et al. Polynomial techniques for investigation of spherical designs. Des. Codes Cryptogr. 51, 275–288 (2009). https://doi.org/10.1007/s10623-008-9260-0
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DOI: https://doi.org/10.1007/s10623-008-9260-0