Publications and Preprints
Nontrivial $t$-designs in polar spaces exist for all $t$
preprint.
arXiv
Abstract
A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space over $\mathbb{F}_p$ equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-$(n,k,\lambda)$ 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 $\lambda$ members of $Y$. Nontrivial examples are currently only known for $t\leq 2$. We show that $t$-$(n,k,\lambda)$ designs in polar spaces exist for all $t$ and $p$ 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 (joint with Kai-Uwe Schmidt)
Combinatorial Theory, 3(1), 2023.
journal arXiv
Abstract
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 (joint with Kai-Uwe Schmidt)
J. Combin. Des. 31(9), 422-431, 2023.
journal arXiv
Abstract
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.
Edited Volumes
Pushing the Limits of Computational Combinatorial Constructions (Dagstuhl Seminar 23161) (L. Moura, A. Nakic, P. Östergård, A. Wassermann, C. Weiß (Editors))
Dagstuhl Reports 13 (2023), no. 4, 40—57.
report
Abstract
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.
Theses
Linear programming bounds in classical association schemes
Dissertation, Paderborn University, Germany, 2023.
PDF
Abstract
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 Delsarte's 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.
Bounds for codes in association schemes via semidefinite programming
Master’s thesis, Otto von Guericke University Magdeburg, Germany, 2017.
Lee sequences and perfect sequences over the quaternions
Bachelor’s thesis, Otto von Guericke University Magdeburg, Germany, 2014.
Talks
(Invited talks are marked with *.)
Existence of $t$-designs in polar spaces for all $t$
Combinatorial Constructions Conference (CCC), Dubrovnik, Croatia, Apr 2024.
* Codes and Steiner Systems *
Oberseminar Groups and Geometry, Bielefeld, Germany, Jan 2024.
* Codes and Designs in Polar Spaces *
On the mathematics of Frédéric Vanhove, Ghent, Belgium, Dec 2023.
Existence of small ordered orthogonal arrays
RICCOTA2023, Rijeka, Croatia, Jul 2023.
Packings and Steiner systems in polar spaces
10th Slovenian Conference on Graph Theory, Kranjska Gora, Slovenia, Jun 2023.
Packings and Steiner systems in polar spaces
Colloquium on Combinatorics, Paderborn, Germany, Nov 2022.
The linear programming bounds in classical association schemes
Finite Geometries—Sixth Irsee Conference, Irsee, Germany, Sept 2022.
(Invitation-only conference)
The linear programming bounds in classical association schemes
Algebraic Coding Theory Summer School—ACT22, Zurich, Switzerland, July 2022.
Packings and Steiner systems in polar spaces
Combinatorics 2022, Mantua, Italy, May 2022.
From Dobble to streaming videos
Science Day of the Faculty of Computer Science, Electrical Engineering and Mathematics, Paderborn, Germany, May 2022.
Won a 1st prize for this talk at the Science Day.
Semidefinite Optimierung und das Packungsproblem der Codierungstheorie
Young Researchers Symposium of the German Mathematical Society DMV, Paderborn, Germany, Mar 2018.
Won an invitation to a workshop at Oberwolfach Research Institute for Mathematics and an award from the company dSpace for this talk.