A quasidouble of the affine plane of order 4 and the solution of a problem on additive designs

A 2-(v,k,λ) block design (P,B) is additive if, up to isomorphism, P can be represented as a subset of a commutative group (G,+) in such a way that the k elements of each block in B sum up to zero in G. If, for some suitable G, the embedding of P in G is also such that, conversely, any zero-sum k-subset of P is a block in B, then (P,B) is said to be strongly additive. In this paper we exhibit the very first examples of additive 2-designs that are not strongly additive, thereby settling an open problem posed in 2019. Our main counterexample is a resolvable 2-(16,4,2) design (F_4×F_4, B_2), which decomposes into two disjoint isomorphic copies of the affine plane of order four. An essential part of our construction is a (cyclic) decomposition of the point-plane design of AG(4,2) into seven disjoint isomorphic copies of the affine plane of order four. This produces, in addition, a solution to Kirkman's schoolgirl problem

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