arXiv subjects
Publications and source records attributed to Jonathan Niemann.
In this article, we describe a new maximal function field $G$ over the finite field $\mathbb{F}_{11^2}$ with $11^2$ elements and show that it cannot be obtained as a subfield of the Hermitian function field. This provides the first known $\mathbb{F}_{p^2}$-maximal function field that satisfies this property. Further, we compute the automorphism group of $G$. The new maximal function field was found using AI.
In this paper we investigate algebraic function fields in positive characteristic mainly obtained as double Artin-Schreier extensions of rational function fields with a plane model. The goal is to extend to such extensions large automorphism groups of the rational function field. In this way, we construct some new families of ordinary function fields and determine their full automorphism groups. Such groups are large with respect to the genus, compared with the known upper bounds on the size of the automorphism group of an ordinary function field.
A $t$-all-symbol PIR code and a $t$-all-symbol batch code of dimension $k$ consist of $n$ servers storing linear combinations of $k$ information symbols with the following recovery property: any symbol stored by a server can be recovered from $t$ pairwise disjoint subsets of servers. In the batch setting, we further require that any multiset of size $t$ of stored symbols can be recovered from~$t$ disjoint subsets of servers. This framework unifies and extends several well-known code families, including one-step majority-logic decodable codes, (functional) PIR codes, and (functional) batch codes. In this paper, we determine the minimum code length for some small values of $k$ and $t$, characterize structural properties of codes attaining this optimum, and derive bounds that show the trade-offs between length, dimension, minimum distance, and $t$. In addition, we study MDS codes and the simplex code, demonstrating how these classical families fit within our framework, and establish new cases of an open conjecture from \cite{YAAKOBI2020} concerning the minimal $t$ for which the simplex code is a $t$-functional batch code.
The classification of maximal function fields over a finite field is a difficult open problem, and even determining isomorphism classes among known function fields is challenging in general. We study a particular family of maximal function fields defined over a finite field with $q^2$ elements, where $q$ is the power of an odd prime. When $d := (q+1)/2$ is a prime, this family is known to contain a large number of non-isomorphic function fields of the same genus and with the same automorphism group. We compute the automorphism group and isomorphism classes also in the case where $d$ is not a prime.