# American Institute of Mathematical Sciences

August  2013, 7(3): 279-292. doi: 10.3934/amc.2013.7.279

## New nonexistence results for spherical designs

 1 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 G.Bonchev str., 1113 Sofia, Bulgaria 2 Faculty of Mathematics and Informatics, Sofia University, 5 James Bourchier blvd, 1164 Sofia, Bulgaria

Received  July 2012 Revised  March 2013 Published  July 2013

New nonexistence results for spherical designs of odd strength and odd cardinality are proved by improvements on previously applied polynomial techniques. This implies new bounds on the designs under consideration either in small dimensions and in certain asymptotic process.
Citation: Peter Boyvalenkov, Maya Stoyanova. New nonexistence results for spherical designs. Advances in Mathematics of Communications, 2013, 7 (3) : 279-292. doi: 10.3934/amc.2013.7.279
##### References:
 [1] M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions,'' Dover, New York, 1965. [2] B. Bajnok, Constructions of spherical 3-designs, Des. Codes Crypt., 21 (2000), 11-18. doi: 10.1023/A:1008367006853. [3] S. Boumova, P. Boyvalenkov and D. Danev, Necessary conditions for existence of some designs in polynomial metric spaces, Europ. J. Combin., 20 (1999), 213-225. doi: 10.1006/eujc.1998.0278. [4] S. Boumova, P. Boyvalenkov and D. Danev, New nonexistence results for spherical designs, in "Constructive Theory of Functions'' (ed. B. Bojanov), Darba, Sofia, (2003), 225-232. [5] S. Boumova, P. Boyvalenkov, H. Kulina and M. Stoyanova, Polynomial techniques for investigation of spherical designs, Des. Codes Crypt., 51 (2009), 275-288. doi: 10.1007/s10623-008-9260-0. [6] S. Boumova, P. Boyvalenkov and M. Stoyanova, A method for proving nonexistence of spherical designs of odd strength and odd cardinality, Probl. Inform. Transm., 45 (2009), 110-123; translated from: Probl. Pered. Inform., 45 (2009), 41-55. doi: 10.1134/S0032946009020033. [7] S. Boumova and D. Danev, On the asymptotic behaviour of a necessary condition for existence of spherical designs, in "Proc. Intern. Workshop ACCT,'' (2002), 54-57. [8] P. Boyvalenkov, D. Danev and S. Nikova, Nonexistence of certain spherical designs of odd strengths and cardinalities, Discr. Comp. Geom., 21 (1999), 143-156. doi: 10.1007/PL00009406. [9] P. Boyvalenkov and M. Stoyanova, A new asymptotic bound of the minimum possible odd cardinality of spherical $(2k-1)$-designs, Discrete Math., 310 (2010), 2170-2175. doi: 10.1016/j.disc.2010.04.007. [10] P. Delsarte, J.-M. Goethals and J. J. Seidel, Spherical codes and designs, Geom. Dedicata, 6 (1977), 363-388. [11] V. I. Levenshtein, Universal bounds for codes and designs, in "Handbook of Coding Theory'' (eds. V. Pless and W.C. Huffman), Elsevier, (1998), 499-648. [12]

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##### References:
 [1] M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions,'' Dover, New York, 1965. [2] B. Bajnok, Constructions of spherical 3-designs, Des. Codes Crypt., 21 (2000), 11-18. doi: 10.1023/A:1008367006853. [3] S. Boumova, P. Boyvalenkov and D. Danev, Necessary conditions for existence of some designs in polynomial metric spaces, Europ. J. Combin., 20 (1999), 213-225. doi: 10.1006/eujc.1998.0278. [4] S. Boumova, P. Boyvalenkov and D. Danev, New nonexistence results for spherical designs, in "Constructive Theory of Functions'' (ed. B. Bojanov), Darba, Sofia, (2003), 225-232. [5] S. Boumova, P. Boyvalenkov, H. Kulina and M. Stoyanova, Polynomial techniques for investigation of spherical designs, Des. Codes Crypt., 51 (2009), 275-288. doi: 10.1007/s10623-008-9260-0. [6] S. Boumova, P. Boyvalenkov and M. Stoyanova, A method for proving nonexistence of spherical designs of odd strength and odd cardinality, Probl. Inform. Transm., 45 (2009), 110-123; translated from: Probl. Pered. Inform., 45 (2009), 41-55. doi: 10.1134/S0032946009020033. [7] S. Boumova and D. Danev, On the asymptotic behaviour of a necessary condition for existence of spherical designs, in "Proc. Intern. Workshop ACCT,'' (2002), 54-57. [8] P. Boyvalenkov, D. Danev and S. Nikova, Nonexistence of certain spherical designs of odd strengths and cardinalities, Discr. Comp. Geom., 21 (1999), 143-156. doi: 10.1007/PL00009406. [9] P. Boyvalenkov and M. Stoyanova, A new asymptotic bound of the minimum possible odd cardinality of spherical $(2k-1)$-designs, Discrete Math., 310 (2010), 2170-2175. doi: 10.1016/j.disc.2010.04.007. [10] P. Delsarte, J.-M. Goethals and J. J. Seidel, Spherical codes and designs, Geom. Dedicata, 6 (1977), 363-388. [11] V. I. Levenshtein, Universal bounds for codes and designs, in "Handbook of Coding Theory'' (eds. V. Pless and W.C. Huffman), Elsevier, (1998), 499-648. [12]
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