On resolvable designs

AbstractA balanced incomplete block design (BIBD) B[k, λ;v] is an arrangement of v elements in blocks of k elements each, such that every pair of elements is contained in exactly λ blocks. A BIBD B[k, 1;v] is called resolvable if the blocks can be partitioned into (v−1)(k−1) families each consisting of v/k mutually disjoint blocks. Ray-Chaudhuri and Wilson [8] proved the existence of resolvable BIBD's B[3, 1; v] for every v≡3 (mod 6). In addition to this result, the existence is proved here of resolvable BIBD's B[4, 1; v] for every v≡4 (mod 12)

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

On uniformly resolvable (C4,K1,3)(C_4,K_{1,3})-designs

In this paper we consider the uniformly resolvable decompositions of the complete graph $K_v$ minus a 1-factor $(K_v − I)$ into subgraphs where each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either 4-cycles or 3-stars

Rastavljivi dizajni

Motivacija za pisanje diplomskog rada bila je Kirkmanov problem 15 učenica. Jedna od interpretacija tog problema je rastavljivi dizajn. Na početku smo se upoznali s dizajnima i iznijeli osnovne rezultate koji vrijede za dizajne. Generalizacija dizajna je u parovima balansirani dizajn (PBD). Na toj incidencijskoj strukturi definiramo pojam rastavljivosti. Jedan nuždan uvjet za postojanje dizajna je Fisherova nejednakost. Dizajni koji dostižu Fisherovu nejednakost su simetrični dizajni. Za rastavljive dizajne vrijedi poboljšanje te nejednakosti, Boseova nejednakost. Nama su posebno zanimljivi dizajni koji dostižu Boseovu nejednakost. To su afino rastavljivi dizajni. Koristeći afino rastavljive dizajne možemo konstruirati simetrične dizajne. Na kraju diplomskog rada povezali smo rastavljive dizajne s još jednom zanimljivom strukturom, ekvidistantnim kodovima.Kirkman’s schoolgirl problem motivated the writing of this graduate work. It is a famous example of a resolvable design. At the beginning we made an introduction to designs and gave some basic results that apply to designs. The incidence structure on which we define the notion of resolution is pairwise balanced design (PBD). One necessary condition for the existence of a design is Fisher’s inequality. Designs that attain the equality in Fisher’s inequality are symmetric designs. If a PBD admits a resolution, a stronger result known as Bose’s inequality holds. We are interested in designs attaining the equality in Bose’s inequality. These are affine resolvable designs. We can construct symmetric designs using affine resolvable designs. In the end we explored the connections between equidistant codes and resolvable designs

A Note on a Family of Resolvable Balanced Incomplete Block Designs

6 pages, 1 article*A Note on a Family of Resolvable Balanced Incomplete Block Designs* (Hedayat, A.; Raktoe, B. L.) 6 page

Resolvable designs with large blocks

Resolvable designs with two blocks per replicate are studied from an optimality perspective. Because in practice the number of replicates is typically less than the number of treatments, arguments can be based on the dual of the information matrix and consequently given in terms of block concurrences. Equalizing block concurrences for given block sizes is often, but not always, the best strategy. Sufficient conditions are established for various strong optimalities and a detailed study of E-optimality is offered, including a characterization of the E-optimal class. Optimal designs are found to correspond to balanced arrays and an affine-like generalization.Comment: Published at http://dx.doi.org/10.1214/009053606000001253 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org