404 research outputs found
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.Comment: 16 page
On the Caratheodory rank of polymatroid bases
In this paper we prove that the Carath\'eodory rank of the set of bases of a
(poly)matroid is upper bounded by the cardinality of the ground set.Comment: 7 page
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.Comment: 12 page
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.Comment: 15 pages, 2 figure
Computing graph gonality is hard
There are several notions of gonality for graphs. The divisorial gonality
dgon(G) of a graph G is the smallest degree of a divisor of positive rank in
the sense of Baker-Norine. The stable gonality sgon(G) of a graph G is the
minimum degree of a finite harmonic morphism from a refinement of G to a tree,
as defined by Cornelissen, Kato and Kool. We show that computing dgon(G) and
sgon(G) are NP-hard by a reduction from the maximum independent set problem and
the vertex cover problem, respectively. Both constructions show that computing
gonality is moreover APX-hard.Comment: The previous version only dealt with hardness of the divisorial
gonality. The current version also shows hardness of stable gonality and
discusses the relation between the two graph parameter
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