arXiv subjects

Mario Osvin Pavčević

Publications and source records attributed to Mario Osvin Pavčević.

5 recordsLinked to original sources

arXiv2025

On higher-dimensional symmetric designs

We study two kinds of generalizations of symmetric block designs to higher dimensions, the so-called $\mathcal{C}$-cubes and $\mathcal{P}$-cubes. For small parameters, all examples up to equivalence are determined by computer calculations. Known properties of automorphisms of symmetric designs are extended to autotopies of $\mathcal{P}$-cubes, while counterexamples are found for $\mathcal{C}$-cubes. An algorithm for the classification of $\mathcal{P}$-cubes with prescribed autotopy groups is developed and used to construct more examples. A bound on the dimension of difference sets for $\mathcal{P}$-cubes is proved and shown to be tight in elementary abelian groups. The construction is generalized to arbitrary groups by introducing regular sets of (anti)automorphisms.

math.CO↗

arXiv2023

Three-dimensional Hadamard matrices of Paley type

We describe a construction of three-dimensional Hadamard matrices of even order $v$ such that $v-1$ is a prime power. The construction covers infinitely many orders for which the existence was previously open.

math.CO↗

arXiv2023

Cubes of symmetric designs

We study $n$-dimensional matrices with $\{0,1\}$-entries ($n$-cubes) such that all their $2$-dimensional slices are incidence matrices of symmetric designs. A known construction of these objects obtained from difference sets is generalized so that the resulting $n$-cubes may have inequivalent slices. For suitable parameters, they can be transformed into $n$-dimensional Hadamard matrices with this property. In contrast, previously known constructions of $n$-dimensional designs all give examples with equivalent slices.

math.CO↗

arXiv2015

Mosaics of Combinatorial Designs

Looking at incidence matrices of $t$-$(v,k,λ)$ designs as $v \times b$ 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 $v \times b$ 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.

math.CO↗

arXiv2014

Intersection numbers for subspace designs

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.

math.CO↗