Abstract:We propose a combination of two known computational methods for the construction of designs with prescribed groups of automorphisms: the Kramer-Mesner method and the method of tactical decompositions. This combined method is used to construct new unitals with parameters 2-(65, 5, 1). q

Intersection numbers for subspace designs are introduced and q-analogs of the Mendelsohn and Köhler equations are given. As an application, we are able to determine the intersection structure of a putative q-analog of the Fano plane for any prime power q. It is shown that its existence implies the existence of a 2-(7, 3, q 4 ) q subspace design. Furthermore, several simplified or alternative proofs concerning intersection numbers of ordinary block designs are discussed.

Abstract. An automorphism group of an incidence structure I induces a tactical decomposition on I. It is well known that tactical decompositions of t-designs satisfy certain necessary conditions which can be expressed as equations in terms of the coefficients of tactical decomposition matrices. In this article we present results obtained for tactical decompositions of q-analogs of t-designs, more precisely, of 2-(v, k, λ 2 ; q) designs. We show that coefficients of tactical decomposition matrices of a design over finite field satisfy an equation system analog to the one known for block designs. Furthermore, taking into consideration specific properties of designs over the binary field F 2 , we obtain an additional system of inequations for these coefficients in that case.

Abstract. Looking at incidence matrices of t -(v, k, λ) designs as v×b matrices with 2 possible entries, each of which indicates incidences of a t -design, we introduce the notion of a c -mosaic of designs, having the same number of points and blocks, as a matrix with c different entries, such that each entry defines incidences of a design. In fact, a v×b matrix is decomposed in c incidence matrices of designs, each denoted by a different colour, hence this decomposition might be seen as a tiling of a matrix with incidence matrices of designs as well. These mosaics have applications in experiment design when considering a simultaneous run of several different experiments. We have constructed infinite series of examples of mosaics and state some probably non-trivial open problems. Furthermore we extend our definition to the case of q -analogues of designs in a meaningful way.

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