A note on the stability number of an orthogonality graph

We consider the orthogonality graph Ω ( 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 Ω ( n ) is α ( Ω ( 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]. 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 Ω ( n ) the latter two bounds are equal to 2 n / n .