# Anton BettenKuwait University | KU · Department of Mathmatics

Anton Betten

PhD

## About

85

Publications

7,625

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697

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Introduction

My research is in Combinatorics, in particular in the field of classifying combinatorial structures.
Editor, Journal of Geometry

Additional affiliations

August 2008 - August 2022

August 2002 - August 2008

March 2001 - March 2002

Education

June 1996 - June 2001

June 1991 - June 1996

September 1989 - June 1991

## Publications

Publications (85)

This paper is a contribution to the classification of parallelisms in three-dimensional projective spaces over small finite fields of order q by computer. The smallest space in which parallelisms have not yet been classified is for q=4.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}...

The classification problem for cubic surfaces with 27 lines is concerned with describing a complete set of the projective equivalence classes of such surfaces. Despite a long history of work, the problem is still open. One approach is to use a coarser equivalence relation based on geometric invariants. The Eckardt point configuration is one such in...

We determine the number of cubic surfaces with 27 lines over a finite field Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {F}}}_q$$\end{document}. This is...

A spread in PG(n,q) is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. A parallelism is uniform if all its spreads are isomorphic. Up to isomorphism, there are three spreads of PG(3,4)-regular, subregular and aregular. Therefore, three types of uniform parallelisms are possible. In this wor...

A convex polyhedron is the convex hull of a finite set of points in \({\mathbb R}^3.\) A triangulation of a convex polyhedron is a decomposition into a finite number of 3-simplices such that any two intersect in a common face or are disjoint. A simplicial dissection is a decomposition into a finite number of 3-simplices such that no two share an in...

We describe a very versatile, fast and useful open source software package to compute combinatorial objects up to isomorphism called Orbiter. We provide an overview of some of the design decisions made during development, and we point out similar software packages. We discuss ways in which combinatorial data can be computed, analyzed and permanentl...

A spread in \(\mathrm{PG}(n,q)\) is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, cryptography, e...

In the 1960s, Hirschfeld embarked on a program to classify cubic surfaces with 27 lines over finite fields. This work is a contribution to this problem. We develop an algorithm to classify surfaces with 27 lines over a finite field using the classical theory of double-sixes. This algorithm is used to classify these surfaces over all fields of order...

In a finite projective plane PG(2, q), a set of k points is called (k, n)-arc if the following two properties hold: Every line intersects in at most n points. There exists a line which intersects in exactly n point. We are interested in determining for each q and each n, the largest value of k for which a (k, n)-arc exists in PG(2, q). If possible,...

We find all 270088 parallelisms of PG(3, 4) invariant under a Baer involution. This addresses one of three possibilities for parallelisms invariant under a group of order two. Since all parallelisms with groups or order larger than two are known, this reduces the number of open cases to two more groups of order two and to parallelisms with trivial...

We discuss an algorithm to search for rainbow cliques in vertex-colored graphs. This algorithm is a generalization of the Bron-Kerbosch algorithm to search for maximal cliques in graphs. As an application, we describe a larger algorithm to classify a certain type of geometric-combinatorial objects called BLT-sets. We report on the classification of...

Many problems in Combinatorics and related fields reduce to the problem of computing orbits of groups acting on finite sets. One of the techniques is known under the name Snakes and Ladders. We offer the alternate name poset classification algorithm. We will describe this technique and compare the performance on example problems.

We present two algorithms to classify cubic surfaces over a finite fields. An implementation in the programming system Orbiter will be described.

Many problems in Combinatorics and related fields reduce to the problem of computing orbits of groups acting on finite sets. One of the techniques is known under the name Snakes and Ladders. We offer the alternate name poset classification algorithm. We will describe this technique and compare the performance on example problems.

We present two algorithms to classify cubic surfaces over a finite fields. An implementation in the programming system Orbiter will be described.

We discuss an algorithm to search for rainbow cliques in vertex-colored graphs. This algorithm is a generalization of the Bron-Kerbosch algorithm to search for maximal cliques in graphs. As an application , we describe a larger algorithm to classify a certain type of geometric-combinatorial objects called BLT-sets. We report on the classification o...

In Hirschfeld (J Austral Math Soc 4(1):83–89, 1964), the existence of the cubic surface which arises from a double-six over the finite field of order four was considered. In Hirschfeld (Rend Mat Appl 26:115–152, 1967), the existence and the properties of the cubic surfaces over the finite fields of odd and even order was discussed and classified ov...

FinInG is a package for computation in Finite Incidence Geometry. It provides users with the basic tools to work in various areas of finite geometry from the realms of projective spaces to the flat lands of generalised polygons. The algebraic power of GAP is exploited, particularly in its facility with matrix and permutation groups.

We discuss dual hyperovals of rank 4 over (Formula presented.). In particular, we classify all such dual hyperovals if the ambient space has dimension 7 or 8. We also determine the bilinear dual hyperovals in the case of an ambient space of dimension 9 or 10. A classification of all dual hyperovals in dimension 9 seems possible in the near future.

Using computer search, we classify the line-spreads in PG(3,q) containing a regulus for q=8 and q=9. Via the Andre, Bruck, Bose connection, this contributes to the classification of translation planes of order 64 and 81.

This conference will highlight such areas as • Algebraic Combinatorics • Finite Geometry • Finite Groups • Group Actions • Algebraic Graph Theory • Coding Theory • Design Theory • Association Schemes • Algorithms • Computations • Combinatorial Objects • Algebraic Structures • Classification and Isomorphism Testing • Applications

Packings of PG(3, q) are closely related to Kirkman's problem of the 15 schoolgirls from 1850 and its generalizations: Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast. The packings of PG(3, 2) give rise to two of seven solutions of Kir...

Using computer, we classify the unitals in the Desarguesian projective plane of order 16. We use computational methods based on analysis involving tactical decompositions to break symmetry, making a computer search feasible. We prove that all unitals in PG(2,16)PG(2,16) are known, namely, up to isomorphism, there are exactly two Buekenhout–Metz uni...

A unital 2-(28, 4, 1) design has 28 points, each block has size 4 and every pair of points is on exactly one block. A blocking set in a design is a subset of the point set with the property that every block intersects the blocking set nontrivially but no block is contained in the blocking set. In this work, we classify the unital 2-(28, 4, 1) desig...

Orbiter is a software package to classify discrete objects such as designs, graphs, codes, and objects from finite geometry. It employs the method of breaking the symmetry to attack difficult problem instances by means of subobjects that serve as a stepping stone. The algorithms combine techniques from Group Theory and from Combinatorics. Orbiter i...

We present three families of multiple blocking sets in Desarguesian projective planes of even characteristic. The first and the third construction apply to any translation hyperoval. The second construction applies to arbitrary hyperovals.

In Finite Geometry, a class of objects known as BLT-sets play an important role. They are points on the Q(4,q) quadric satisfying a condition on triples. This paper is a contribution to the difficult problem of classifying these sets up to isomorphism, i.e., up to the action of the automorphism group of the quadric. We reduce the classification pro...

It is known that 42 is the largest size of a 6-arc in the Desargue-sian projective plane of order 8. In this paper, we classify these (42, 6)8 arcs. Equivalently, we classify the smallest 3-fold blocking sets in PG(2, 8), which have size 31.

The 157, 211 triangle-free symmetric configurations are classified and some of their
properties are examined. We conclude that each such configuration has a blocking set. Further
properties like transitivity on lines, self-duality, and self-polarity are discussed.

The paper summarises existing theory and classifications for finite line-transitive linear spaces, develops the theory further,
and organises it in a way that enables its effective application. The starting point is a theorem of Camina and the fifth
author that identifies three kinds of line-transitive automorphism groups of linear spaces. In two o...

We present a class of transitive BLT-sets that contains several known examples and at least one new example over the field with 41 elements with automorphism group ℤ 21 :(ℤ 2 ×ℤ 2 ). This example is sporadic in the sense that it does not fall into any of the known infinite families of BLT-sets.

A linear space is Drake / Larson if it contains at least two lines and there are no lines of size 2,3 or 6. The existence or nonexistence of such linear spaces on v points is known except for v = 30. The purpose of this paper is to settle the remaining case on thirty points in the negative. This result relies on a combination of parameter calculati...

We present two families of constacyclic linear codes with large automorphism groups. The codes are obtained from the twisted tensor product construction. AMS subject classification: 05E20, 05B25, 11T71, 94B25, 94B27, 51E22, 51E20, 20G40, 14L35

For a fixed integer k, all but a finite number of line-transitive linear spaces with lines of size k are point-primitive. In this paper, we study the finite class of examples where a line-transitive group is point-imprimitive, that is, preserves a non-trivial partition of the point set. We restrict to the case where (i) the number of unordered poin...

In 1991, Weidong Fang and Huiling Li proved that there are only finitely many
non-trivial linear spaces that admit a line-transitive, point-imprimitive group
action, for a given value of gcd(k,r), where k is the line size and r is the
number of lines on a point. The aim of this paper is to make that result
effective. We obtain a classification of a...

Rota's conjecture states that the number of minimal excluded minors for the class of GF(q)-representable matroids is finite. The conjecture holds for q = 2, 3, and 4, but remains unresolved for fields of order 5 and higher. At present only six 7-element minimal excluded minors for GF(5)-representation arc known. We found two 8-element, nine 9-eleme...

A parallel version of an algorithm for solving systems of integer linear equations with {0,1}-variables is presented. The algorithm is based on lattice basis reduction in combination with explicit enumeration.

This text offers a thorough introduction to the mathematical concepts behind the theory of error-correcting linear codes. Care is taken to introduce the necessary algebraic concepts, for instance the theory of finite fields, the polynomial rings over such fields and the ubiquitous concept of group actions that allows the classification of codes by...

In particular, a cubic graph on n vertices (n even) corresponds to a regular linear space of type (3n/2∣ 3ⁿ, 2c), where c = 3n(3n− 10)/8, and conversely. 6.23 Remark Let Kn denote the complete graph on n vertices (which is regular of degree n−1). Then Constructions 6.19 and 6.22 give regular linear spaces of type (equation found). 6.24 Construction...

We extend the enumeration of regular linear spaces in 1 to at most 19 points. In addition, one of the 5 missing cases in the previous list is settled. The number of regular linear spaces of type (15|215,330) is 10,177,328. © 2005 Wiley Periodicals, Inc. J Combin Designs.

We present an algorithm that constructs and classifies finite linear spaces admitting a line-transitive group which leaves invariant a nontrivial point-partition. The algorithm was developed from an algorithm of Nickel and Niemeyer that classified such linear spaces on 729 points with line size 8, but was never published. It has been applied to com...

We address the problem of constructing hyperovals and arcs in De-sarguesian projective planes of even order. Our purpose is twofold. On the one hand, we show the connection to geometric codes, which are the codes spanned by the characteristic vectors of subspaces of a fixed dimension in projective spaces. Using these codes, we deduce a new hyperova...

We present an algorithm that constructs and classifies finite linear spaces admitting a given line-transitive group which leaves invariant a nontrivial point-partition. This algorithm was developed from an algorithm of Nickel and Niemeyer that classified such linear spaces on 729 points with line-size 8, but was never published. It has been applied...

Recent years have seen a dramatic increase in existence results for t-designs with large t, i.e. t ≥ 5. Designs are now known to exist for several thousand parameter sets, mostly constructed by the method of orbiting under a group. This note is a contribution to the classification of these designs by parameters. We take an abstract look at admissib...

A program is outlined for the enumeration of unital 2-(28,4,1) designs that uses tactical decompositions defined by vectors of certain weight in the dual binary code of a design. A class of designs with a spread that covers a codeword of weight 12 is studied in detail. A total of 909 nonisomorphic designs are constructed that include the classical...

Let G be a finite primitive permutation group with a non-trivial, non-regular normal subgroup N, and let G be an orbit of a point stabilizer Na. Then each composition factor S of Na occurs as a section of the permutation group induced by Na on G. The case N ¼ G is a theorem of Wielandt. The general result and some of its corollaries are useful for...

In [4] we constructed and enumerated all proper linear spaces on 17 points using the so-called TDO-method. This method is also strong enough for the construction and enumeration of all proper linear spaces on 18 points. In the present note we list the results. We get 2412890 proper linear spaces on 18 points. AMS subject classification: 05B25, 05B3...

A new simple 6-(14,7,4) design is presented with automorphism group isomorphic to A 4 . Combining the derived and the residual designs of the 6-(14,7,4) designs, which were previously known, the extension method of van Leijenhorst and Tran Van Trung results in a large number of simple 5-(14,7,18) designs with trivial automorphism group. This parame...

In this article we give tables of configurations v3 for v⩽18 and triangle-free configurations for v⩽21 together with some statistics about some properties of the structures like transitivity, self-duality or self-polarity.

The first 5-(72, 6, 1) designs with automorphism group PSL(2, 71) were found by Mills [10]. We presently enumerate all 5-(72, 6, 1) designs with this automorphism group. There are in all 926299 non-isomorphic designs. We show that a necessary condition for semiregular5-(v, 6, 1) designs with automorphism group PSL(2, v 1) to exist is thatv=84, 228...

Rahilly families of pre-difference sets have been introduced by Rahilly, Praeger, Street and Bryant as a tool for constructing symmetric designs. Using orderly generation, we construct Rahilly families for various groups up to equivalence. For each equivalence class we determine the isomorphism type of the corresponding design. Some designs may be...

The topic of this paper is to determine the isomorphism types of designs which are invariant under a given group. As an example, we consider SQS(20) invariant under a subgroup of the symmetric group Ë ¾¼ isomorphic to the alternating group .

The study of configurations or — more generally — finite incidence geometries is best accomplished by taking into account also their automorphism groups. These groups act on the geometry and in particular on points, blocks, flags and even anti-flags. The orbits of these groups lead to tactical decompositions of the incidence matrices of the geometr...

Up to now, all known Steiner 5-designs are on q + 1 points where q ≡ 3 (mod 4) is a prime power and the design is admitting PSL(2, q) as a group of automorphisms. In this article we present a 5-(36,6,1) design admitting PGL(2, 17) × C
2 as a group of automorphisms. The design is unique with this automorphism group and even for the commutator group...

In this short note, simple 8-(40,11,1440) designs with automorphism group PSL(4,3) are presented. The designs are constructed with the method of Kramer and Mesner on a computer using the software package DISCRETA (Betten et al., A tool for constructing t-designs, Lehrstuhl II für Mathematik, Universität Bayreuth. Software package and documentation...

The proper linear spaces on 17 points are classified. The computation is based on the parameters of the geometries and makes extensive use of tactical decompositions. A specific one, the tactical decomposition by ordering (TDO) which has been invented by Betten and Braun (Combinatorics’90, Ann. Discrete Math., Vol. 52, North-Holland, Amsterdam, 199...

. We give several examples where PVM was successfully used as a tool for distributed computation of solutions of combinatorial problems. Involved topics are the computation of the solution of multidimensional subset sum problems which appear in the construction of block designs, the construction and classification of finite solvable groups up to is...

Kramer-Mesner matrices have been used as a powerful tool to construct t-designs. In this paper we construct Kramer-Mesner matrices for fixed values of k and t in which the entries are polynomials in n the number of vertices of the underlying graph. From this we obtain an elementary proof that with a few exceptions Sn[2] is a maximal subgroup of or...

Recent results in the search for simple t-designs and large sets of t-designs are reported. Many new parameter sets of simple t-designs on up to 40 points and t ⩾ 7 are given. The tool used is a program DISCRETA, developed by the authors, which applies the method of Kramer-Mesner (Discrete Math. 15 (1976) 263–296) where an automorphism group of the...

We describe a computer search for the construction of simple designs with prescribed automorphism groups. Using our program package DISCRETA this search yields designs with parameter sets 7-(33, 8, 10), 7-(27, 9, 60), 7-(26, 9, λ) for λ = 54, 63, 81, 7-(26, 8, 6), 7-(25, 9, λ) for λ = 45, 54, 72, 7-(24, 9, λ) for λ = 40, 48, 64, 7-(24, 8, λ) for λ...

The 28,872,973 linear spaces on 12 points are constructed. The parameters of the geometries play an important role. In order to make generation easy, we construct possible parameter sets for geometries first (purely algebraically). Afterwards, the corresponding geometries are tried to construct. We define line types, point types, point cases, and a...

. We show the existence of simple 8-(31,10,93) and 8-(31,10,100) designs. For each value of we show 3 designs in full detail. The designs are constructed with a prescribed group of automorphisms PSL(3; 5) using the method of Kramer and Mesner [8]. They are the first 8designs with small parameters which are known explicitly. We do not yet know if PS...

Eine Einführung in die Theorie der linearen Codes, in der zyklische Codes besonders ausführlich behandelt werden. Großer Wert wird auch auf computerunterstützte Methoden gelegt, insbesondere für die Bestimmung der Minimaldistanz linearer Codes, für die Abzählung der Isometrieklassen linearer Codes sowie Blockcodes und für die Erzeugung von Repräsen...

Abstract We present a new approach to the construction of simple block de- signs. Using the computer package DISCRETA, we start with the con- struction of block designs which are invariant with respect to some pre- scribed group of automorphisms. Therefore, one applies the method of Kramer and Mesner which means that one has to solve systems of dio...

. We call a finite linear space regular, if all pencils of lines are similar. This means that the way how the lines through a point partition the complement of this point is equivalent for all points. We enumerate all finite regular linear spaces of order 14 and, with some gaps, up to order 16. We comment on some of these spaces, point out interrel...

Some simple 7-designs with small parameters are constructed with the aid of a computer. The smallest parameter set found is 7-(24; 8; 4): An automorphism group is prescribed for finding the designs and used for determining the isomorphism types. Further designs are derived from these designs by known construction processes.

A computer package is being developed at Bayreuth for the generation and investigation of discrete structures. The package is a C and C++ class library of powerful algorithms endowed with graphical interface modules. Standard applications can be run automatically whereas research projects mostly require small C or C++ programs. The basic philosophy...

An algorithm for the construction of finite solvable groups of small order is given. A parallelized version under PVM is presented.

Recent results in the search for simple t-designs are reported. There are 31 parameter sets of simple 7-designs and many parameter sets of new simple 6-designs on up to 33 points listed up. The tool used is a program DISCRETA, developed by the authors, which applies the method of Kramer-Mesner 6] where an automorphism group of the desired designs i...

A computer package is being developed at Bayreuth for the generation and investigation of discrete structures. The package is a C and C++ class library of powerful algorithms endowed with graphical interface modules. Standard applications can be run automatically whereas research projects mostly require small C or C++ programs. The basic philosophy...