Nontrivial $t$-designs in polar spaces exist for all $t$ - Weiß, Charlene (Des. Codes Cryptogr.)
A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space over $\mathbb{F}_q$ equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-$(n,k,λ)$ design in a finite classical polar space of rank $n$ is a collection $Y$ of totally isotropic $k$-spaces such that each totally isotropic $t$-space is contained in exactly $λ$ members of $Y$. Nontrivial examples are currently only known for $t≤ 2$. We show that $t$-$(n,k,λ)$ designs in polar spaces exist for all $t$ and $q$ provided that $k>\frac{21}{2}t$ and $n$ is sufficiently large enough. The proof is based on a probabilistic method by Kuperberg, Lovett, and Peled, and it is thus nonconstructive.

Packings and Steiner systems in polar spaces - Schmidt, Kai-Uwe; Weiß, Charlene (Combinatorial Theory)
A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-Steiner system in a finite classical polar space of rank $n$ is a collection $Y$ of totally isotropic $n$-spaces such that each totally isotropic $t$-space is contained in exactly one member of $Y$. Nontrivial examples are known only for $t=1$ and $t=n-1$. We give an almost complete classification of such $t$-Steiner systems, showing that such objects can only exist in some corner cases. This classification result arises from a more general result on packings in polar spaces.

Existence of small ordered orthogonal arrays - Schmidt, Kai‐Uwe; Weiß, Charlene (Journal of Combinatorial Designs)
We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled.

Linear programming bounds in classical association schemes - Weiß, Charlene
Digital communications relies heavily on the usage of different types of codes. Prominent codes nowadays are rank-metric codes and subspace codes - the q-analogs of binary codes and binary codes with constant weight. All these codes can be viewed as subsets of classical association schemes. A central coding-theoretic problem is to derive upper bounds for the size of codes. This thesis investigates Delsartes powerful linear program whose optimum is precisely such a bound for codes in association schemes. The linear programs for binary codes and binary constant-weight codes have been extensively studied since the 1970s, but their optimum is still unknown. We determine in a unified way the optimum of the linear program in several ordinary q-analogs as well as in their affine counterparts. In particular, bounds and constructions for codes in polar spaces are established, where the bounds are sharp up to a constant factor in many cases. Moreover, based on these results, an almost complete classification of Steiner systems in polar spaces is provided by showing that they could only exist in some corner cases.

Pushing the Limits of Computational Combinatorial Constructions (Dagstuhl Seminar 23161) - Moura, Lucia; Nakic, Anamari; Östergård, Patric; Wassermann, Alfred; Weiß, Charlene
This report documents the program and the outcomes of Dagstuhl Seminar 23161 "Pushing the Limits of Computational Combinatorial Constructions". In this Dagstuhl Seminar, we focused on computational methods for challenging problems in combinatorial construction. This includes algorithms for construction of combinatorial objects with prescribed symmetry, for isomorph-free exhaustive generation, and for combinatorial search. Examples of specific algorithmic techniques are tactical decomposition, the Kramer-Mesner method, algebraic methods, graph isomorphism software, isomorph-free generation, clique-finding methods, heuristic search, SAT solvers, and combinatorial optimization. There was an emphasis on problems involving graphs, designs and codes, also including topics in related fields such as finite geometry, graph decomposition, Hadamard matrices, Latin squares, and q-analogs of designs and codes.
Keywords: automorphism groups; combinatorial algorithms; finite geometries; subspace designs