Searcharxiv⌕ Search

arXiv subjects

Dion Gijswijt

Publications and source records attributed to Dion Gijswijt.

25 records · Page 2Linked to original sources
arXiv2012

Violating the Shannon capacity of metric graphs with entanglement

The Shannon capacity of a graph G is the maximum asymptotic rate at which messages can be sent with zero probability of error through a noisy channel with confusability graph G. This extensively studied graph parameter disregards the fact that on atomic scales, Nature behaves in line with quantum mechanics. Entanglement, arguably the most counterintuitive feature of the theory, turns out to be a useful resource for communication across noisy channels. Recently, Leung, Mancinska, Matthews, Ozols and Roy [Comm. Math. Phys. 311, 2012] presented two examples of graphs whose Shannon capacity is strictly less than the capacity attainable if the sender and receiver have entangled quantum systems. Here we give new, possibly infinite, families of graphs for which the entangled capacity exceeds the Shannon capacity.

quant-ph↗
arXiv2011

Characterizing partition functions of the vertex model

We characterize which graph parameters are partition functions of a vertex model over an algebraically closed field of characteristic 0 (in the sense of de la Harpe and Jones). We moreover characterize when the vertex model can be taken so that its moment matrix has finite rank.

math.CO↗
arXiv2010

Matrix Algebras and Semidefinite Programming Techniques for Codes

This PhD thesis is concerned with SDP bounds for codes: upper bounds for (non)-binary error correcting codes and lower bounds for (non)-binary covering codes. The methods are based on the method of Schrijver that uses triple distances in stead of pairs as in the classical Delsarte bound. The main topics discussed are: 1) Block-diagonalisation of matrix *-algebras, 2) Terwilliger-algebra of the nonbinary Hamming scheme (including an explicit block-diagonalisation), 3) SDP-bounds for (nonbinary) error-correcting codes and covering codes (including computational results), 4) Discussion on the relation with matrix-cuts, 5) Computational results for Affine caps.

math.CO↗
arXiv2010

Polyhedra with the Integer Caratheodory Property

A polyhedron P has the Integer Caratheodory Property if the following holds. For any positive integer k and any integer vector w in kP, there exist affinely independent integer vectors x_1,...,x_t in P and positive integers n_1,...,n_t such that n_1+...+n_t=k and w=n_1x_1+...+n_tx_t. In this paper we prove that if P is a (poly)matroid base polytope or if P is defined by a TU matrix, then P and projections of P satisfy the integer Caratheodory property.

math.CO↗
arXiv2009

Block diagonalization for algebra's associated with block codes

For a matrix *-algebra B, consider the matrix *-algebra A consisting of the symmetric tensors in the n-fold tensor product of B. Examples of such algebras in coding theory include the Bose-Mesner algebra and Terwilliger algebra of the (non)binary Hamming cube, and algebras arising in SDP-hierarchies for coding bounds using moment matrices. We give a computationally efficient block diagonalization of A in terms of a given block diagonalization of B, and work out some examples, including the Terwilliger algebra of the binary- and nonbinary Hamming cube. As a tool we use some basic facts about representations of the symmetric group.

math.OC↗
arXiv2005

On a set-theoretic invariant

Let a_1,...,a_m be positive real numbers. Besser and Moree considered weighted numbers of -1,+1 solutions of the linear inequality |a_i-a_j| < e_ka_k < a_i+a_j, with e_k=-1 of 1 and k running over the integers 1,...,m with i and j skipped. They introduced some invariants and near invariants related to this situation (invariant meaning here: not depending on the choice of i and j). The main result of their paper is extended here to a much more general setting, namely that of certain maps from finite sets to {-1,1}. Some applications are given.

math.CO↗