A note on the stability number of an orthogonality graph

  • Authors:

  • E. de Klerk;D. V. Pasechnik

  • Affiliations:

  • Department of Econometrics and Operations Research, Faculty of Economics and Business Administration, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands;School of Physical and Mathematical Sciences, Nanyang Technological University, 1 Nanyang Walk, Blk 5, Singapore 637616, Singapore

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2007

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Abstract

We consider the orthogonality graph @W(n) with 2^n vertices corresponding to the vectors {0,1}^n, two vertices adjacent if and only if the Hamming distance between them is n/2. We show that, for n=16, the stability number of @W(n) is @a(@W(16))=2304, thus proving a conjecture of V. Galliard [Classical pseudo telepathy and coloring graphs, Diploma Thesis, ETH Zurich, 2001. Available at http://math.galliard.ch/Cryptography/Papers/PseudoTelepathy/SimulationOfEntanglement.pdf]. The main tool we employ is a recent semidefinite programming relaxation for minimal distance binary codes due to A. Schrijver [New code upper bounds from the Terwilliger algebra, IEEE Trans. Inform. Theory 51 (8) (2005) 2859-2866]. Also, we give a general condition for a Delsarte bound on the (co)cliques in graphs of relations of association schemes to coincide with the ratio bound, and use it to show that for @W(n) the latter two bounds are equal to 2^n/n.