## 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.

PDF 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.

PDF 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.

## Theses

**Linear programming bounds in classical association schemes**

PhD thesis, 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

*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.