arXiv subjects

Alfred Wassermann

Publications and source records attributed to Alfred Wassermann.

At least 19 recordsLinked to original sources

arXiv2025

$\mathrm{ EA}(q)$-additive Steiner 2-designs

A design is $G$-additive with $G$ an abelian group, if its points are in $G$ and each block is zero-sum in $G$. All the few known ``manageable" additive Steiner 2-designs are $\mathrm{EA}(q)$-additive for a suitable $q$, where $\mathrm{EA}(q)$ is the elementary abelian group of order $q$. We present some general constructions for $\mathrm{EA}(q)$-additive Steiner 2-designs which unify the known ones and allow to find a few new ones: an additive $\mathrm{EA}(2^8)$-additive 2-$(52,4,1)$ design which is also resolvable, and three pairwise non-isomorphic $\mathrm{EA}(3^5)$-additive 2-$(121,4,1)$ designs, none of which is the point-line design of $\mathrm{PG}(4,3)$. In the attempt to find also an $\mathrm{EA}(2^9)$-additive 2-$(511,7,1)$ design, we prove that a putative 2-analog of a 2-$(9,3,1)$ design cannot be cyclic.

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

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

Solving the Market Split Problem with Lattice Enumeration

The market split problem was proposed by Cornu\'ejols and Dawande in 1998 as benchmark problem for algorithms solving linear systems with binary variables. The recent (2025) Quantum Optimization Benchmark Library (QOBLIB) contains a set of feasible instances of the market split problem. In QOBLIB an instance of the market split problem is considered as solved as soon as at least one feasible solution has been found. The market split problem seems to be difficult to solve with the conventional branch-and-cut approach of integer linear programming software which reportedly can handle QOBLIB instances up to $m=7$. In contrast, a new GPU implementation of the Schroeppel-Shamir algorithm solves instances up to $m=11$. In this note we report about experiments with an algorithm that reduces the market split problem to a lattice problem. With the author's most recent implementation - named solvediophant - instances of the QOBLIB market split benchmark problems can be solved up to $m=14$ on a standard computer.

math.OC↗

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↗

arXiv2024

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↗

arXiv2024

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↗

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↗

arXiv2020

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 $k > \theta_1 > \theta_2 > \theta_3$, and that the triple $(\theta_1, \theta_2, \theta_3)$ 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↗

arXiv2020

Graph decompositions in projective geometries

Let PG$(\mathbb{F}_q^v)$ be the $(v-1)$-dimensional projective space over $\mathbb{F}_q$ and let $Γ$ be a simple graph of order ${q^k-1\over q-1}$ for some $k$. A 2$-(v,Γ,λ)$ design over $\mathbb{F}_q$ is a collection $\cal B$ of graphs (\textit{blocks}) isomorphic to $Γ$ with the following properties: the vertex set of every block is a subspace of PG$(\mathbb{F}_q^v)$; every two distinct points of PG$(\mathbb{F}_q^v)$ are adjacent in exactly $λ$ blocks. This new definition covers, in particular, the well known concept of a 2$-(v,k,λ)$ design over $\mathbb{F}_q$ corresponding to the case that $Γ$ is complete. In this work of a foundational nature we illustrate how difference methods allow us to get concrete non-trivial examples of $Γ$-decompositions over $\mathbb{F}_2$ or $\mathbb{F}_3$ for which $Γ$ is a cycle, a path, a prism, a generalized Petersen graph, or a Moebius ladder. In particular, we will discuss in detail the special and very hard case that $Γ$ is complete and $λ=1$, i.e., the Steiner 2-designs over a finite field. Also, we briefly touch the new topic of near resolvable 2-$(v,2,1)$ designs over $\mathbb{F}_q$. This study has led us to some (probably new) collateral problems concerning difference sets. Supported by multiple examples, we conjecture the existence of infinite families of $Γ$-decompositions over a finite field that can be obtained by suitably labeling the vertices of $Γ$ with the elements of a Singer difference set.

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

Majority-logic Decoding with Subspace Designs

Rudolph (1967) introduced one-step majority logic decoding for linear codes derived from combinatorial designs. The decoder is easily realizable in hardware and requires that the dual code has to contain the blocks of so called geometric designs as codewords. Peterson and Weldon (1972) extended Rudolphs algorithm to a two-step majority logic decoder correcting the same number of errors than Reed's celebrated multi-step majority logic decoder. Here, we study the codes from subspace designs. It turns out that these codes have the same majority logic decoding capability as the codes from geometric designs, but their majority logic decoding complexity is sometimes drastically improved.

math.CO↗

arXiv2019

A subspace code of size $333$ in the setting of a binary $q$-analog of the Fano plane

We show that there is a binary subspace code of constant dimension 3 in ambient dimension 7, having minimum distance 4 and cardinality 333, i.e., $333 \le A_2(7,4;3)$, which improves the previous best known lower bound of 329. Moreover, if a code with these parameters has at least 333 elements, its automorphism group is in one of $31$ conjugacy classes. This is achieved by a more general technique for an exhaustive search in a finite group that does not depend on the enumeration of all subgroups.

math.CO↗

arXiv2019

$q$-analogs of group divisible designs

A well known class of objects in combinatorial design theory are {group divisible designs}. Here, we introduce the $q$-analogs of group divisible designs. It turns out that there are interesting connections to scattered subspaces, $q$-Steiner systems, design packings and $q^r$-divisible projective sets. We give necessary conditions for the existence of $q$-analogs of group divsible designs, construct an infinite series of examples, and provide further existence results with the help of a computer search. One example is a $(6,3,2,2)_2$ group divisible design over $\operatorname{GF}(2)$ which is a design packing consisting of $180$ blocks that such every $2$-dimensional subspace in $\operatorname{GF}(2)^6$ is covered at most twice.

math.CO↗

arXiv2018

The lengths of projective triply-even binary codes

It is shown that there does not exist a binary projective triply-even code of length $59$. This settles the last open length for projective triply-even binary codes. Therefore, projective triply-even binary codes exist precisely for lengths $15$, $16$, $30$, $31$, $32$, $45$--$51$, and $\ge 60$.

math.CO↗

arXiv2018

Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6

The maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=6$, and constant dimension $k=4$ is $257$, where the $2$ isomorphism types are extended lifted maximum rank distance codes. In finite geometry terms the maximum number of solids in $\operatorname{PG}(7,2)$, mutually intersecting in at most a point, is $257$. The result was obtained by combining the classification of substructures with integer linear programming techniques. This implies that the maximum size $A_2(8,6)$ of a binary mixed-dimension code of packet length $8$ and minimum subspace distance $6$ is $257$ as well.

math.CO↗