A note on the stability number of an orthogonality graph.

Abstract

We consider the orthogonality graph Ω(n) with 2n 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 by 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 Schrijver [New code upper bounds from the Terwilliger algebra, IEEE Trans. Inform. Theory 51 (8)

(2005) 2859–2866]. As well, we give a general condition for Delsarte bound on the (co)cli¬ques 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 2n/n.