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

New nonexistence results for spherical designs

Peter Boyvalenkov 1, and  Maya Stoyanova 2,

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.
Keywords: Spherical designs, bounds for spherical designs..
Mathematics Subject Classification: Primary: 05B30; Secondary: 94B6.
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. Google Scholar

[2]

B. Bajnok, Constructions of spherical 3-designs, Des. Codes Crypt., 21 (2000), 11-18. doi: 10.1023/A:1008367006853.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Delsarte, J.-M. Goethals and J. J. Seidel, Spherical codes and designs, Geom. Dedicata, 6 (1977), 363-388.  Google Scholar

[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.  Google Scholar

[12]

, ., , ().   Google Scholar

show all references

References:
[1]

M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions,'' Dover, New York, 1965. Google Scholar

[2]

B. Bajnok, Constructions of spherical 3-designs, Des. Codes Crypt., 21 (2000), 11-18. doi: 10.1023/A:1008367006853.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Delsarte, J.-M. Goethals and J. J. Seidel, Spherical codes and designs, Geom. Dedicata, 6 (1977), 363-388.  Google Scholar

[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.  Google Scholar

[12]

, ., , ().   Google Scholar

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