12 research outputs found
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
Parallel Class Intersection Matrices of Orthogonal Resolutions
This work was partially supported by the Bulgarian National Science Fund under Contract
No MM 1405. Part of the results were announced at the Fifth International Workshop on Optimal Codes
and Related Topics (OCRT), White Lagoon, June 2007, BulgariaParallel class intersection matrices (PCIMs) have been defined
and used in [6], [14], [15] for the classification of resolvable designs with several
parameter sets. Resolutions which have orthogonal resolutions (RORs)
have been classified in [19] for designs with some small parameters. The
present paper deals with the additional restrictions that the existence of an
orthogonal mate might impose on the PCIMs of a resolution, and with the
effect of both PCIMs usage and the methods for RORs construction described in
[19] and [20]. It is shown in several examples how consideration of
PCIMs can result in constructing only of solutions which can have orthogonal mates,
and thus substantially improve the computation time. There are
parameters for which PCIMs make the classification of RORs possible, and
also cases when PCIMs directly prove the nonexistence of doubly resolvable
designs with certain parameters
Orthogonal Resolutions and Latin Squares
Resolutions which are orthogonal to at least one other resolution (RORs) and sets of m mutually orthogonal resolutions (m-MORs) of 2-(v, k, λ) designs are considered. A dependence of the number of nonisomorphic RORs and m-MORs of multiple designs on the number of inequivalent sets of v/k − 1 mutually orthogonal latin squares (MOLS) of size m is obtained. ACM Computing Classification System (1998): G.2.1.∗ This work was partially supported by the Bulgarian National Science Fund under Contract
No I01/0003
Resolvability of infinite designs
In this paper we examine the resolvability of infinite designs. We show that in stark contrast to the finite case, resolvability for infinite designs is fairly commonplace. We prove that every t-(v,k,Λ) design with t finite, v infinite and k,λ<v is resolvable and, in fact, has α orthogonal resolutions for each α<v. We also show that, while a t-(v,k,Λ) design with t and λ finite, v infinite and k=v may or may not have a resolution, any resolution of such a design must have v parallel classes containing v blocks and at most λ−1 parallel classes containing fewer than v blocks. Further, a resolution into parallel classes of any specified sizes obeying these conditions is realisable in some design. When k<v and λ=v and when k=v and λ is infinite, we give various examples of resolvable and non-resolvable t-(v,k,Λ) designs
University of Montana Commencement Program, 1977
Commencement program from the University of Montana.https://scholarworks.umt.edu/um_commencement_programs/1079/thumbnail.jp