A note on the stability number of an orthogonality graph

We consider the orthogonality graph Omega(n) with 2^n vertices corresponding to the 0-1 n-vectors, two vertices adjacent if and only if the Hamming distance between them is n/2. We show that the stability number of Omega(16) is alpha(Omega(16))= 2304, thus proving a conjecture by Galliard. The main tool we employ is a recent semidefinite programming relaxation for minimal distance binary codes due to Schrijver. As well, we give a general condition for 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 Omega(n) the latter two bounds are equal to 2^n/n.Comment: 10 pages, LaTeX, 1 figure, companion Matlab code. Misc. misprints fixed and references update

A Note on the Stability Number of an Orthogonality Graph

We consider the orthogonality graph (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 (n) is ( (16)) = 2304, thus proving a conjecture by Galliard [7].The main tool we employ is a recent semidefinite programming relaxation for minimal distance binary codes due to Schrijver [16].Moreover, we give a general condition for 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 (n) the latter two bounds are equal to 2n/n.C0;C61

The road to deterministic matrices with the restricted isometry property

The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.Comment: 24 page

Lecture notes: Semidefinite programs and harmonic analysis

Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands.Comment: 31 page

The Independence Number of the Orthogonality Graph in Dimension 2k2^k

We determine the independence number of the orthogonality graph on $2^k$-dimensional hypercubes. This answers a question by Galliard from 2001 which is motivated by a problem in quantum information theory. Our method is a modification of a rank argument due to Frankl who showed the analogous result for $4p^k$-dimensional hypercubes, where $p$ is an odd prime.Comment: 3 pages, accepted by Combinatorica, fixed a minor typo spotted by Peter Si

Massive MIMO for Crowd Scenarios: A Solution Based on Random Access

This paper presents a new approach to intra-cell pilot contamination in crowded massive MIMO scenarios. The approach relies on two essential properties of a massive MIMO system, namely near-orthogonality between user channels and near-stability of channel powers. Signal processing techniques that take advantage of these properties allow us to view a set of contaminated pilot signals as a graph code on which iterative belief propagation can be performed. This makes it possible to decontaminate pilot signals and increase the throughput of the system. The proposed solution exhibits high performance with large improvements over the conventional method. The improvements come at the price of an increased error rate, although this effect is shown to decrease significantly for increasing number of antennas at the base station