8,833 research outputs found
Covering of Subspaces by Subspaces
Lower and upper bounds on the size of a covering of subspaces in the
Grassmann graph \cG_q(n,r) by subspaces from the Grassmann graph
\cG_q(n,k), , are discussed. The problem is of interest from four
points of view: coding theory, combinatorial designs, -analogs, and
projective geometry. In particular we examine coverings based on lifted maximum
rank distance codes, combined with spreads and a recursive construction. New
constructions are given for with or . We discuss the density
for some of these coverings. Tables for the best known coverings, for and
, are presented. We present some questions concerning
possible constructions of new coverings of smaller size.Comment: arXiv admin note: text overlap with arXiv:0805.352
Constructions of biangular tight frames and their relationships with equiangular tight frames
We study several interesting examples of Biangular Tight Frames (BTFs) -
basis-like sets of unit vectors admitting exactly two distinct frame angles
(ie, pairwise absolute inner products) - and examine their relationships with
Equiangular Tight Frames (ETFs) - basis-like systems which admit exactly one
frame angle.
We demonstrate a smooth parametrization BTFs, where the corresponding frame
angles transform smoothly with the parameter, which "passes through" an ETF
answers two questions regarding the rigidity of BTFs. We also develop a general
framework of so-called harmonic BTFs and Steiner BTFs - which includes the
equiangular cases, surprisingly, the development of this framework leads to a
connection with the famous open problem(s) regarding the existence of Mersenne
and Fermat primes. Finally, we construct a (chordally) biangular tight set of
subspaces (ie, a tight fusion frame) which "Pl\"ucker embeds" into an ETF.Comment: 19 page
Resolving sets for Johnson and Kneser graphs
A set of vertices in a graph is a {\em resolving set} for if, for
any two vertices , there exists such that the distances . In this paper, we consider the Johnson graphs and Kneser
graphs , and obtain various constructions of resolving sets for these
graphs. As well as general constructions, we show that various interesting
combinatorial objects can be used to obtain resolving sets in these graphs,
including (for Johnson graphs) projective planes and symmetric designs, as well
as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems
and toroidal grids.Comment: 23 pages, 2 figures, 1 tabl
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
Universal Entanglers for Bosonic and Fermionic Systems
A universal entangler (UE) is a unitary operation which maps all pure product
states to entangled states. It is known that for a bipartite system of
particles with a Hilbert space ,
a UE exists when and . It is also
known that whenever a UE exists, almost all unitaries are UEs; however to
verify whether a given unitary is a UE is very difficult since solving a
quadratic system of equations is NP-hard in general. This work examines the
existence and construction of UEs of bipartite bosonic/fermionic systems whose
wave functions sit in the symmetric/antisymmetric subspace of
. The development of a theory of UEs for
these types of systems needs considerably different approaches from that used
for UEs of distinguishable systems. This is because the general entanglement of
identical particle systems cannot be discussed in the usual way due to the
effect of (anti)-symmetrization which introduces "pseudo entanglement" that is
inaccessible in practice. We show that, unlike the distinguishable particle
case, UEs exist for bosonic/fermionic systems with Hilbert spaces which are
symmetric (resp. antisymmetric) subspaces of
if and only if (resp. ). To prove this we employ algebraic geometry to reason about the different
algebraic structures of the bosonic/fermionic systems. Additionally, due to the
relatively simple coherent state form of unentangled bosonic states, we are
able to give the explicit constructions of two bosonic UEs. Our investigation
provides insight into the entanglement properties of systems of
indisitinguishable particles, and in particular underscores the difference
between the entanglement structures of bosonic, fermionic and distinguishable
particle systems.Comment: 15 pages, comments welcome, TQC2013 Accepted Tal
On the number of k-dominating independent sets
We study the existence and the number of -dominating independent sets in
certain graph families. While the case namely the case of maximal
independent sets - which is originated from Erd\H{o}s and Moser - is widely
investigated, much less is known in general. In this paper we settle the
question for trees and prove that the maximum number of -dominating
independent sets in -vertex graphs is between and
if , moreover the maximum number of
-dominating independent sets in -vertex graphs is between
and . Graph constructions containing a large number of
-dominating independent sets are coming from product graphs, complete
bipartite graphs and with finite geometries. The product graph construction is
associated with the number of certain MDS codes.Comment: 13 page
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