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Michael Kiermaier

Publications and source records attributed to Michael Kiermaier.

At least 19 recordsLinked to original sources
arXiv2026

Intersection numbers for designs in regular semilattices

We generalize intersection numbers for combinatorial designs to designs in finite meet-semilattices satisfying suitable regularity conditions. While designs in regular semilattices go back to Delsarte, our regularity assumptions are weaker than his and need not give rise to an association scheme. In this framework, we extend Mendelsohn's equations, prove a generalized Singleton bound with Steiner systems as equality cases, and determine the block intersection distribution at any block of a Steiner system. In particular, this distribution is independent of the chosen block. Specializing to several classical semilattice families, our results recover a number of well-known distributions in coding and design theory. In the Hamming and the $q$-Hamming (or bilinear forms) schemes, they give the local distance distributions of MDS and MRD codes, respectively. In the Johnson and $q$-Johnson (or Gra{\ss}mann) schemes, they reproduce the block intersection distribution of classical and $q$-analog Steiner systems, equivalently the distance distribution of diameter-perfect constant-weight codes and diameter-perfect constant-dimension subspace codes. For the $q$-Johnson schemes, to the best of our knowledge, this result is new. As a further illustration, we apply our theory to designs of perfect matchings. Our approach provides a unified treatment of these cases in the strongest form known in the literature, determining the distribution relative to each individual block or codeword, without averaging and without linearity or additivity assumptions. Moreover, it identifies the natural double-counting objects underlying these distributions, leading to formulas in the regularity parameters of the semilattice and avoiding the more cumbersome expressions that arise in eigenvalue-based approaches via the ambient association scheme.

math.CO↗
arXiv2025

The paired construction for Boolean functions on the slice

Let $V$ be a finite set of size $n$. We consider real functions on the "slice" $\binom{V}{k}$, which are also known as functions in the Johnson scheme. For $I \subseteq J \subseteq V$, the characteristic function of the set of all $K\in\binom{V}{k}$ with $I \subseteq K \subseteq J$ is called "basic". In this article, we investigate a construction arising as the sum of two "opposite" basic functions. In essentially all cases, these "paired" functions are Boolean. Our main result is the determination of the exact degree -- regarding a representation by an $n$-variable polynomial -- of all paired functions. The proof is elementary and does not involve any spectral methods. First, we settle the middle layer case $n=2k$ by identifying and combining various relations among the degrees involved. Then the general case is reduced to the middle layer situation by means of derived, reduced, and dual functions. Remarkably, in certain situations, the degree is strictly smaller than what is guaranteed by the elementary upper bound for the sum of functions. This makes paired functions good candidates for fixed-degree Boolean functions of small support size. As it turns out, for $n = 2k$ and even degree $t \notin \{0,k\}$, paired functions provide the smallest known non-zero Boolean functions, surpassing the $t$-pencils, which is the smallest known construction in all other cases.

math.CO↗
arXiv2025

Designs in finite classical polar spaces

Combinatorial designs have been studied for nearly 200 years. Fifty years ago, Cameron, Delsarte, and Ray-Chaudhury started investigating their q-analogs, also known as subspace designs or designs over finite fields. Designs can be defined analogously in finite classical polar spaces, too. The definition includes the m-regular systems from projective geometry as the special case where the blocks are generators of the polar space. The first nontrivial such designs for t > 1 were found by De Bruyn and Vanhove in 2012, and some more designs appeared recently in the PhD thesis of Lansdown. In this article, we investigate the theory of classical and subspace designs for applicability to designs in polar spaces, explicitly allowing arbitrary block dimensions. In this way, we obtain divisibility conditions on the parameters, derived and residual designs, intersection numbers and an analog of Fisher's inequality. We classify the parameters of symmetric designs. Furthermore, we conduct a computer search to construct designs of strength t=2, resulting in designs for more than 140 previously unknown parameter sets in various classical polar spaces over GF(2) and GF(3).

math.CO↗
arXiv2025

The degree of functions in the Johnson and q-Johnson schemes

In 1982, Cameron and Liebler investigated certain "special sets of lines" in PG(3,q), and gave several equivalent characterizations. Due to their interesting geometric and algebraic properties, these "Cameron-Liebler line classes" got much attention. Several generalizations and variants have been considered in the literature, the main directions being a variation of the dimensions of the involved spaces, and studying the analogous situation in the subset lattice. An important tool is the interpretation of the objects as Boolean functions in the "Johnson" and "q-Johnson schemes". In this article, we develop a unified theory covering all these variations. Generalized versions of algebraic and geometric properties will be investigated, having a parallel in the notion of "designs" and "antidesigns" in association schemes, which is connected to Delsarte's concept of "design-orthogonality". This leads to a natural definition of the "degree" and the "weights" of functions in the ambient scheme, refining the existing definitions. We will study the effect of dualization and of elementary modifications of the ambient space on the degree and the weights. Moreover, a divisibility property of the sizes of Boolean functions of degree t will be proven.

math.CO↗
arXiv2025

Steiner 3-designs as extensions

In this article, we construct a Steiner system with the parameters $S(3,6,42)$, settling one of the smallest open parameter sets of Steiner $3$-designs. Furthermore, we establish the existence of rotational Steiner quadruple systems on $46$ and $92$ points. Our construction method is based on extending Steiner $2$-designs using prescribed extension groups. We also consider extensions to designs of higher strength. The article includes a table and a discussion of the status of all admissible parameters for Steiner $3$-designs on at most $50$ points.

math.CO↗
arXiv2025

Varieties of Nodal surfaces, coding theory and Discriminants of cubic hypersurfaces. Part 1: Generalities and nodal K3 surfaces. Part 2: Cubic Hypersurfaces, associated discriminants. Part 3: Nodal quintics. Part 4: Nodal sextics

We attach two binary codes to a projective nodal surface (the strict code K and, for even degree d, the extended code K' ) to investigate the `Nodal Severi varieties F(d, n) of nodal surfaces in P^3 of degree d and with n nodes, and their incidence hierarchy, relating partial smoothings to code shortenings. Our first main result solves a question which dates back over 100 years: the irreducible components of F(4, n) are in bijection with the isomorphism classes of their extended codes K', and these are exactly all the 34 possible shortenings of the extended Kummer code K' , and a component is in the closure of another if and only if the code of the latter is a shortening of the code of the former. We extend this result classifying the irreducible components of all nodal K3 surfaces in the same way, and we fully classify their extended codes. In this classification there are some sporadic cases, obtain through projection from a node. For surfaces of degree d=5 in P^3 we determine (with one possible exception) all the possible codes K, and for several cases of K, we show the irreducibility of the corresponding open set of F(5, n), for instance we show the irreducibility of the family of Togliatti quintic surfaces. In the fourth part we show that a `Togliatti-like' description holds for surfaces of degree 6 with the maximum number of nodes= 65: they are discriminants of cubic hypersurfaces in P^6 with 31 (respectively 32) nodes, and we have an irreducible 18-dimensional family of them. For degree d=6, our main result is based on some novel auxiliary results: 1) the study of the half-even sets of nodes on sextic surfaces, 2) the investigation of discriminants of cubic hypersurfaces X, 3) the computer assisted proof that, for n = 65, both codes K, K' are uniquely determined, 4) the description of these codes, relating the geometry of the Barth sextic with the Doro-Hall graph.

math.AG↗
arXiv2025

Some new Steiner designs $S(2,6,91)$

The Kramer-Mesner method for constructing designs with a prescribed automorphism group $G$ has proven effective many times. In the special case of Steiner designs, the task reduces to solving an exact cover problem, with the advantage that fast backtracking solvers like Donald Knuth's dancing links and dancing cells can be used. We find ways to encode the inherent symmetry of the problem space, induced by the action of the normalizer of $G$, into a single instance of the exact cover problem. This eliminates redundant computations of certain isomorphic search branches, while preventing the overhead caused by repeatedly restarting the solver. Our improved approach is applied to the parameters $S(2,6,91)$. Previously, only four such Steiner designs were known, all of which had been constructed as cyclic designs over four decades ago. We find $23$ new designs, each with full automorphism group of order $84$.

math.CO↗
arXiv2023

Higher incidence matrices and tactical decomposition matrices

In 1985, Janko and Tran Van Trung published an algorithm for constructing symmetric designs with prescribed automorphisms. This algorithm is based on the equations by Dembowski (1958) for tactical decompositions of point-block incidence matrices. In the sequel, the algorithm has been generalized and improved in many articles. In parallel, higher incidence matrices have been introduced by Wilson in 1982. They have proven useful for obtaining several restrictions on the existence of designs. For example, a short proof of the generalized Fisher's inequality makes use of these incidence matrices. In this paper, we introduce a unified approach to tactical decompositions and incidence matrices. It works for both combinatorial and subspace designs alike. As a result, we obtain a generalized Fisher's inequality for tactical decompositions of combinatorial and subspace designs. Moreover, our approach is explored for the construction of combinatorial and subspace designs of arbitrary strength.

math.CO↗
arXiv2023

Classification of $Δ$-divisible linear codes spanned by codewords of weight $Δ$

We classify all $q$-ary $Δ$-divisible linear codes which are spanned by codewords of weight $Δ$. The basic building blocks are the simplex codes, and for $q=2$ additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight $4$ have been classified, which is the case $q=2$ and $Δ=4$ of our classification. As an application, we give an alternative proof of a theorem of Liu on binary $Δ$-divisible codes of length $4Δ$ in the projective case.

math.CO↗
arXiv2022

On strongly walk regular graphs, triple sum sets and their codes

Strongly walk regular graphs (SWRGs or $s$-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length~2 are replaced by paths of length~$s$. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an $s$-SWRG has at most 4 distinct eigenvalues $k > θ_1 > θ_2 > θ_3$, and that the triple $(θ_1, θ_2, θ_3)$ satisfies a certain homogeneous polynomial equation of degree $s - 2$ (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for $s = 5$ and $s = 7$, there are only the obvious solutions, and we conjecture this to remain true for all (odd) $s \ge 9$.

math.CO↗
arXiv2022

Linear codes associated with the Desarguesian ovoids in $Q^+(7,q)$

The Desarguesian ovoids in the orthogonal polar space $Q^+(7,q)$ with $q$ even have first been introduced by Kantor by examining the $8$-dimensional absolutely irreducible modular representations of $\text{PGL}(2,q^3)$. We investigate this module for all prime power values of $q$. The shortest $\text{PGL}(2,q^3)$-orbit $O$ gives the Desarguesian ovoid in $Q^+(7,q)$ for even $q$ and it is known to give a complete partial ovoid of the symplectic polar space $W(7,q)$ for odd~$q$. We determine the hyperplane sections of $O$. As a corollary, we obtain the parameters $[q^3+1,8,q^3-q^2-q]_q$ and the weight distribution of the associated $\mathbb{F}_q$-linear code $C_O$ and the parameters $[q^3+1,q^3-7,5]_q$ of the dual code $C_O^\perp$ for $q \ge 4$. We also show that both codes $C_O$ and $C_O^\perp$ are length-optimal for all prime power values of $q$.

cs.IT↗
arXiv2021

On $α$-points of $q$-analogs of the Fano plane

Arguably, the most important open problem in the theory of $q$-analogs of designs is the question for the existence of a $q$-analog $D$ of the Fano plane. It is undecided for every single prime power value $q \geq 2$. A point $P$ is called an $α$-point of $D$ if the derived design of $D$ in $P$ is a geometric spread. In 1996, Simon Thomas has shown that there must always exist at least one non-$α$-point. For the binary case $q = 2$, Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non-$α$-points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of $α$-points implies the existence of a partiton of the symplectic generalized quadrangle $W(q)$ into spreads. As a consequence, the statement of Heden and Sissokho is generalized to all primes $q$ and all even values of $q$.

math.CO↗
arXiv2020

On the lengths of divisible codes

In this article, the effective lengths of all $q^r$-divisible linear codes over $\mathbb{F}_q$ with a non-negative integer $r$ are determined. For that purpose, the $S_q(r)$-adic expansion of an integer $n$ is introduced. It is shown that there exists a $q^r$-divisible $\mathbb{F}_q$-linear code of effective length $n$ if and only if the leading coefficient of the $S_q(r)$-adic expansion of $n$ is non-negative. Furthermore, the maximum weight of a $q^r$-divisible code of effective length $n$ is at most $σq^r$, where $σ$ denotes the cross-sum of the $S_q(r)$-adic expansion of $n$. This result has applications in Galois geometries. A recent theorem of N{ă}stase and Sissokho on the maximum size of a partial spread follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension subspace codes.

math.CO↗
arXiv2019

Tables of subspace codes

One of the main problems of subspace coding asks for the maximum possible cardinality of a subspace code with minimum distance at least $d$ over $\mathbb{F}_q^n$, where the dimensions of the codewords, which are vector spaces, are contained in $K\subseteq\{0,1,\dots,n\}$. In the special case of $K=\{k\}$ one speaks of constant dimension codes. Since this (still) emerging field is very prosperous on the one hand side and there are a lot of connections to classical objects from Galois geometry it is a bit difficult to keep or to obtain an overview about the current state of knowledge. To this end we have implemented an on-line database of the (at least to us) known results at \url{subspacecodes.uni-bayreuth.de}. The aim of this recurrently updated technical report is to provide a user guide how this technical tool can be used in research projects and to describe the so far implemented theoretic and algorithmic knowledge.

math.CO↗
arXiv2019

On projective $q^r$-divisible codes

A projective linear code over $\mathbb{F}_q$ is called $Δ$-divisible if all weights of its codewords are divisible by $Δ$. Especially, $q^r$-divisible projective linear codes, where $r$ is some integer, arise in many applications of collections of subspaces in $\mathbb{F}_q^v$. One example are upper bounds on the cardinality of partial spreads. Here we survey the known results on the possible lengths of projective $q^r$-divisible linear codes.

math.CO↗
arXiv2019

On the minimum number of minimal codewords

We study the minimum number of minimal codewords in linear codes from the point of view of projective geometry. We derive bounds and in some cases determine the exact values. We also present an extension to minimal subcode supports.

math.CO↗
arXiv2019

Three-weight codes over rings and strongly walk regular graphs

We construct strongly walk-regular graphs as coset graphs of the duals of codes with three non-zero homogeneous weights over $\mathbb{Z}_{p^m},$ for $p$ a prime, and more generally over chain rings of depth $m$, and with a residue field of size $q$, a prime power. Infinite families of examples are built from Kerdock and generalized Teichmüller codes. As a byproduct, we give an alternative proof that the Kerdock code is nonlinear.

math.CO↗