582 research outputs found
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
A Method for Classification of Doubly Resolvable Designs and Its Application
This article presents the principal results of the Ph.D. thesis Investigation and classification of doubly resolvable designs by Stela Zhelezova (Institute of Mathematics and Informatics, BAS), successfully defended at the Specialized Academic Council for Informatics and Mathematical
Modeling on 22 February 2010.The resolvability of combinatorial designs is intensively investigated because of its applications. This research focuses on resolvable designs
with an additional property - they have resolutions which are mutually orthogonal. Such designs are called doubly resolvable. Their specific properties can be used in statistical and cryptographic applications.Therefore the classification of doubly resolvable designs and their sets of mutually orthogonal resolutions might be very important. We develop a method for classification of doubly resolvable designs. Using this method and extending it with some theoretical restrictions we succeed in obtaining a classification of doubly resolvable designs with small parameters. Also we classify 1-parallelisms and 2-parallelisms of PG(5,2) with automorphisms of order 31 and find the first known transitive 2-parallelisms among them. The content of the paper comprises the essentials of the author’s Ph.D. thesis
Frequency permutation arrays
Motivated by recent interest in permutation arrays, we introduce and
investigate the more general concept of frequency permutation arrays (FPAs). An
FPA of length n=m lambda and distance d is a set T of multipermutations on a
multiset of m symbols, each repeated with frequency lambda, such that the
Hamming distance between any distinct x,y in T is at least d. Such arrays have
potential applications in powerline communication. In this paper, we establish
basic properties of FPAs, and provide direct constructions for FPAs using a
range of combinatorial objects, including polynomials over finite fields,
combinatorial designs, and codes. We also provide recursive constructions, and
give bounds for the maximum size of such arrays.Comment: To appear in Journal of Combinatorial Design
Graphical Designs and Gale Duality
A graphical design is a subset of graph vertices such that the weighted
averages of certain graph eigenvectors over the design agree with their global
averages. We use Gale duality to show that positively weighted graphical
designs in regular graphs are in bijection with the faces of a generalized
eigenpolytope of the graph. This connection can be used to organize, compute
and optimize designs. We illustrate the power of this tool on three families of
Cayley graphs -- cocktail party graphs, cycles, and graphs of hypercubes -- by
computing or bounding the smallest designs that average all but the last
eigenspace in frequency order. We also prove that unless NP = coNP, there
cannot be an efficient description of all minimal designs that average a fixed
number of eigenspaces in a graph.Comment: 30 pages, 14 figures, 1 tabl
Equivalence of Decoupling Schemes and Orthogonal Arrays
We consider the problem of switching off unwanted interactions in a given
multi-partite Hamiltonian. This is known to be an important primitive in
quantum information processing and several schemes have been presented in the
literature to achieve this task. A method to construct decoupling schemes for
quantum systems of pairwise interacting qubits was introduced by M.
Stollsteimer and G. Mahler and is based on orthogonal arrays. Another approach
based on triples of Hadamard matrices that are closed under pointwise
multiplication was proposed by D. Leung. In this paper, we show that both
methods lead to the same class of decoupling schemes. Moreover, we establish a
characterization of orthogonal arrays by showing that they are equivalent to
decoupling schemes which allow a refinement into equidistant time-slots.
Furthermore, we show that decoupling schemes for networks of higher-dimensional
quantum systems with t-local Hamiltonians can be constructed from classical
error-correcting codes.Comment: 26 pages, latex, 1 figure in tex
Constructing packings in Grassmannian manifolds via alternating projection
This paper describes a numerical method for finding good packings in
Grassmannian manifolds equipped with various metrics. This investigation also
encompasses packing in projective spaces. In each case, producing a good
packing is equivalent to constructing a matrix that has certain structural and
spectral properties. By alternately enforcing the structural condition and then
the spectral condition, it is often possible to reach a matrix that satisfies
both. One may then extract a packing from this matrix.
This approach is both powerful and versatile. In cases where experiments have
been performed, the alternating projection method yields packings that compete
with the best packings recorded. It also extends to problems that have not been
studied numerically. For example, it can be used to produce packings of
subspaces in real and complex Grassmannian spaces equipped with the
Fubini--Study distance; these packings are valuable in wireless communications.
One can prove that some of the novel configurations constructed by the
algorithm have packing diameters that are nearly optimal.Comment: 41 pages, 7 tables, 4 figure
- …