300,766 research outputs found
Continuous Association Schemes and Hypergroups
Classical finite association schemes lead to a finite-dimensional algebras
which are generated by finitely many stochastic matrices. Moreover, there exist
associated finite hypergroups. The notion of classical discrete association
schemes can be easily extended to the possibly infinite case. Moreover, the
notion of association schemes can be relaxed slightly by using suitably
deformed families of stochastic matrices by skipping the integrality
conditions. This leads to larger class of examples which are again associated
to discrete hypergroups.
In this paper we propose a topological generalization of the notion of
association schemes by using a locally compact basis space and a family of
Markov-kernels on indexed by a further locally compact space where the
supports of the associated probability measures satisfy some partition
property. These objects, called continuous association schemes, will be related
to hypergroup structures on . We study some basic results for this new
notion and present several classes of examples. It turns out that for a given
commutative hypergroup the existence of an associated continuous association
scheme implies that the hypergroup has many features of a double coset
hypergroup. We in particular show that commutative hypergroups, which are
associated with commutative continuous association schemes, carry dual positive
product formulas for the characters. On the other hand, we prove some rigidity
results in particular in the compact case which say that for given spaces
there are only a few continuous association schemes
Matroids and Quantum Secret Sharing Schemes
A secret sharing scheme is a cryptographic protocol to distribute a secret
state in an encoded form among a group of players such that only authorized
subsets of the players can reconstruct the secret. Classically, efficient
secret sharing schemes have been shown to be induced by matroids. Furthermore,
access structures of such schemes can be characterized by an excluded minor
relation. No such relations are known for quantum secret sharing schemes. In
this paper we take the first steps toward a matroidal characterization of
quantum secret sharing schemes. In addition to providing a new perspective on
quantum secret sharing schemes, this characterization has important benefits.
While previous work has shown how to construct quantum secret sharing schemes
for general access structures, these schemes are not claimed to be efficient.
In this context the present results prove to be useful; they enable us to
construct efficient quantum secret sharing schemes for many general access
structures. More precisely, we show that an identically self-dual matroid that
is representable over a finite field induces a pure state quantum secret
sharing scheme with information rate one
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
Coherent configurations and triply regular association schemes obtained from spherical designs
Delsarte-Goethals-Seidel showed that if is a spherical -design with
degree satisfying , carries the structure of an association
scheme. Also Bannai-Bannai showed that the same conclusion holds if is an
antipodal spherical -design with degree satisfying . As a
generalization of these results, we prove that a union of spherical designs
with a certain property carries the structure of a coherent configuration. We
derive triple regularity of tight spherical -designs, mutually unbiased
bases, linked symmetric designs with certain parameters.Comment: 17page
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