Alfred Wassermann

• University of Bayreuth

The celebrated Kramer-Mesner method for the construction of designs with a prescribed automorphism group $G$ has been 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 ut...

Combinatorial designs have been studied for nearly 200 years. 50 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 projectiv...

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 in...

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 in...

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 paramete...

The classification of unitals with parameters 2-(28, 4, 1) according to the 2-rank of their incidence matrices was initiated by McGuire, Tonchev and Ward, who proved that the 2-rank of any unital on 28 points is greater than or equal to 19, and up to isomorphism, there is a unique unital with 2-rank equal to 19. Jaffe and Tonchev investigated the n...

The classification of unitals with parameters 2-(28, 4, 1) according to the 2-rank of their incidence matrices was initiated by McGuire, Tonchev and Ward, who proved that the 2-rank of any unital on 28 points is greater than or equal to 19, and up to isomorphism, there is a unique unital with 2-rank equal to 19. Jaffe and Tonchev investigated the n...

In 1986, Kreher and Radziszowski were the first who used the famous LLL algorithm to construct combinatorial designs. Subsequently, lattice algorithms have been applied to construct a large variety of objects in design theory, coding theory and finite geometry. Unfortunately, the use of lattice algorithms in combinatorial search is still not well e...

A recent article in this Journal (J. Chem. Educ. 2020, 97, 1715-1730) contains erroneous isomer numbers of certain sets of organic compounds. We show that correct numbers are produced by constructive enumeration of isomers using a recent molecular structure generator.

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 para...

Let PG ( F q v ) be the ( v − 1 ) ‐dimensional projective space over F q and let Γ be a simple graph of order q k − 1 q − 1 for some k . A 2 − ( v , Γ , λ ) design over F q is a collection ℬ of graphs (blocks) isomorphic to Γ with the following properties: the vertex set of every block is a subspace of PG ( F q v ) ; every two distinct points of PG...

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 Rudolph’s algorithm to a two-step majority logic decod...

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.

A projective linear code over $\mathbb{F}_q$ is called $\Delta$-divisible if all weights of its codewords are divisible by $\Delta$. 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 sp...

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

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, packing designs and qr-divisible projective sets. We give necessary conditions for the existence of q-an...

Rudoph (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. Here, we study the codes from subspace designs. It turns out that these codes have the same...

Let PG$(\mathbb{F}_q^v)$ be the $(v-1)$-dimensional projective space over $\mathbb{F}_q$ and let $\Gamma$ be a simple graph of order ${q^k-1\over q-1}$ for some $k$. A 2$-(v,\Gamma,\lambda)$ design over $\mathbb{F}_q$ is a collection $\cal B$ of graphs (blocks) isomorphic to $\Gamma$ with the following properties: the vertex-set of every block is a...

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...

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 $\ge 60$.

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 existenc...

We report the computer construction of 1316 mutually disjoint 2-(13, 3, 1)2 subspace designs. By combining disjoint designs and using supplementary subspace designs we conclude that subspace designs exist for 1≤λ≤2047.

In this article, we show the existence of large sets $\operatorname{LS}_2[3](2,k,v)$ for infinitely many values of $k$ and $v$. The exact condition is $v \geq 8$ and $0 \leq k \leq v$ such that for the remainders $\bar{v}$ and $\bar{k}$ of $v$ and $k$ modulo $6$ we have $2 \leq \bar{v} < \bar{k} \leq 5$. The proof is constructive and consists of tw...

For discrete structures which are based on a finite ambient set and its subsets there exists the notion of a “q-analog”: For this, the ambient set is replaced by a finite vector space and the subsets are replaced by subspaces. Consequently, cardinalities of subsets become dimensions of subspaces. Subspace designs are the q-analogs of combinatorial...

Subspace designs are the q-analogs of combinatorial designs. Introduced in the 1970s, these structures gained a lot of interest recently because of their application to random network coding. Compared to combinatorial designs, the number of blocks of subspace designs are huge even for the smallest instances. Thus, for a computational approach, soph...

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...

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 cla...

For which positive integers $n,k,r$ does there exist a linear $[n,k]$ code $C$ over $\mathbb{F}_q$ with all codeword weights divisible by $q^r$ and such that the columns of a generating matrix of $C$ are projectively distinct? The motivation for studying this problem comes from the theory of partial spreads, or subspace codes with the highest possi...

Let denote the maximum cardinality of a set of k-dimensional subspaces of an n-dimensional vector space over the finite field of order q, , such that any two different subspaces have a distance of at least d. Lower bounds on can be obtained by explicitly constructing corresponding sets . When searching for such sets with a prescribed group of auto...

Using integer linear programming and table-lookups, we prove that there is no binary linear [1988, 12, 992] code. As a by-product, the non-existence of binary linear codes with the parameters [324, 10, 160], [356, 10, 176], [772, 11, 384], and [836, 11, 416] is shown. Our work is motivated by the recent construction of the extended dualized Kerdock...

The question if there exists a q-analog of the Fano plane is open since it was first posed in 1972. For a putative binary q-analog of the Fano plane all automorphisms of order greater than 4 had been excluded previously. Here, it is shown with theoretical and computational methods that the order of the automorphism group of a binary q-analog of the...

It is shown that the automorphism group of a binary $q$-analog of the Fano plane is either trivial or of order $2$.

In this article, we show the existence of large sets $\operatorname{LS}_2[3](2,k,v)$ for infinitely many values of $k$ and $v$. The exact condition is $v \geq 8$ and $0 \leq k \leq v$ such that for the remainders $\bar{v}$ and $\bar{k}$ of $v$ and $k$ modulo $6$ we have $2 \leq \bar{v} < \bar{k} \leq 5$. The proof is constructive and consists of tw...

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.

We prove that the automorphism group of an extremal binary self-dual code does not contain elements of order . Combining this with the known results in the literature, one obtains that divides for a non-negative integer a.

The main problem 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 e...

MOLGEN 5.x combines the efficiency of the molecular generator MOLGEN 3.5 and the flexibility of MOLGEN 4.x. To achieve this, the software was reimplemented based on a totally new concept. The most visible new features are fuzzy molecular formula input and explicit use of atom state patterns. We describe the version MOLGEN 5.0 of this new series. ©...

Using integer linear programming and table-lookups we prove that there is no binary linear [1988,12,992] code. As a byproduct, the non-existence of binary linear [324,10,160], [356,10,176], [772,11,384], and [836,11,416] codes is shown. On the other hand, there exists a linear (994,4^6,992) code over Z_4. Its Gray image is a binary non-linear (1988...

In this paper we investigate codes over finite commutative rings R, whose generator matrices are built from \$\alpha\$-circulant matrices. For a non-trivial ideal I<R we give a method to lift such codes over R/I to codes over R, such that some isomorphic copies are avoided. For the case where I is the minimal ideal of a finite chain ring we refine...

Based on results in finite geometry we prove the existence of MRD codes in (F_q)_(n,n) with minimum distance $n$ which are essentially different from Gabidulin codes. The construction results from algebraic structures which are closely related to those of finite fields. Furthermore we show that an analogue of MacWilliams' extension theorem does not...

In this eBook we introduce our readers to one of the most comprehensive and thematically diverse treatise on the emerging discipline of mathematical chemistry, or, more accurately, discrete mathematical chemistry. Although mathematical representation and characterization of chemical objects were known for a long time, the incursion of discrete math...

A $t\text{-}(n,k,\lambda;q)$-design is a set of $k$-subspaces, called blocks, of an $n$-dimensional vector space $V$ over the finite field with $q$ elements such that each $t$-subspace is contained in exactly $\lambda$ blocks. A partition of the complete set of $k$-subspaces of $V$ into disjoint $t\text{-}(n,k,\lambda;q)$ designs is called a large...

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...

MOLGEN 5.x combines the efficiency of the molecular generator MOLGEN 3.5 and the flexibility of MOLGEN 4.x. To achieve this, the software was reimplemented based on a totally new concept. The most visible new features are fuzzy molecular formula input and explicit use of atom state patterns. We here describe the first version MOLGEN 5.0 of this new...

In this paper, we classify quadratic and cubic self-dual bent functions in eight variables with the help of computers. There are exactly four and 45 non-equivalent self-dual bent functions of degree two and three, respectively. This result is achieved by enumerating all eigenvectors with ± 1 entries of the Sylvester Hadamard matrix with an integer...

Let $\F_q^n$ be a vector space of dimension $n$ over the finite field $\F_q$. A $q$-analog of a Steiner system (briefly, a $q$-Steiner system), denoted $S_q[t,k,n]$, is a set $S$ of $k$-dimensional subspaces of $\F_q^n$ such that each $t$-dimensional subspace of $\F_q^n$ is contained in exactly one element of $S$. Presently, $q$-Steiner systems are...

In this paper we give the first construction of a q-analog of a Steiner system. Using a computer search we found at least 26 q-Steiner Systems S_2[2,3,13] admitting the normalizer of a singer cycle as a group of automorphisms.

A fast method to compute the minimum Lee weight and the symmetrized weight enumerator of extended quadratic residue codes (XQR-codes) over the ring $\BBZ_{4}$ is developed. Our approach is based on the classical Brouwer–Zimmermann algorithm and additionally takes advantage of the large group of automorphisms and the self-duality of the $\BBZ_{4}$ -...

We construct linear codes over finite fields with prescribed minimum distance by selectiong columns of the generator matrix. This selection problem can be formulated as an integer programming problem. In order to reduce the search space we prescribe a group of automorphisms. Then, in many cases the resulting integer programming problem can be solve...

Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the mi...

We describe the integration of a well known algorithm for computing and displaying plane loci based on ideal elimination using Gröbner bases in the dynamic geometry software JSXGraph. With our approach it is not only possible to determine loci depending on other loci but it is also possible to extend JSXGraph to deal with loci depending on arbitrar...

The article gives constructions of disjoint 5-designs obtained from permutation groups and extremal self-dual codes. Several new simple 5-designs are found with parameters that were left open in the table of 5-designs given in [G. B. Khosrovshahi and R. Laue, t-designs with t≥3, in: “Handbook of Combinatorial Designs”, 2nd edn., C. J. Colbourn and...

Based on ideas of K\"otter and Kschischang we use constant dimension subspaces as codewords in a network. We show a connection to the theory of q-analogues of a combinatorial designs, which has been studied in Braun, Kerber and Laue as a purely combinatorial object. For the construction of network codes we successfully modified methods (constructio...

In this paper, we construct new binary singly even self-dual codes with larger minimum weights than the previously known singly even self-dual codes for several lengths. Several known construction methods are used to construct the new self-dual codes.

Although forbidden in democratic states a market for votes could improve efficiency. This paper examines the way trading votes will change the outcome of an election. A two-party model with two phases is presented. In the pre-trading phase the two parties set their agendas consisting of a direct subsidy and an income tax. In the trading phase voter...

We construct new binary and ternary self-orthogonal linear codes. In order to do this we use an equivalence between the existence of a self-orthogonal linear code with a prescribed minimum distance and the existence of a solution of a certain system of Diophantine linear equations. To reduce the size of the system of equations we restrict the searc...

The program system MOLGEN is devoted to the generation of all structural formulae (= connectivity isomers) that correspond to a given molecular formula, optionally under further constraints, e.g. the necessary presence or absence of substructures. It arose from the idea to provide an efficient and portable tool for molecular structure generation an...

The program system MOLGEN is devoted to generating all structures (connectivity isomers, constitutions) that correspond to a given molecular formula, with optional further restrictions, e.g. presence or absence of particular substructures. MOLGEN arose from the idea to provide an efficient and portable tool for molecular structure elucidation in ch...

The purpose of this study was to determine the effect of below-knee compression stockings on running performance in men runners. Using a within-group study design, 21 moderately trained athletes (39.3 +/- 10.9 years) without lower-leg abnormities were randomly assigned to perform a stepwise treadmill test up to a voluntary maximum with and without...

The only example of a binary doubly-even self-dual [120,60,20] code was found in 2005 by P. Gaborit, C.-S. Nedeloaia and A. Wassermann [On the weight enumerators of duadic and quadratic residue codes. IEEE Trans. Inform. Theory 51, 402–407 (2005)]. In this work we present 25 new binary doubly-even self-dual [120,60,20] codes having an automorphism...

The class of the quadratic residue codes (QR-codes) over the ring Zopf4 contains very good Zopf4-linear codes. It is well known that the Gray images of the QR-codes over Zopf4 of length 8, 32 and 48 are non-linear binary codes of higher minimum Hamming distance than comparable known linear codes. The QR-Code of length 48 is also the largest one who...

We show the existence of simple 8-(31,12,3080), 8-(40,12,16200) and 8-(40,12,16520) designs and list for each parameter set one example in full detail. The designs are constructed with a prescribed group of automorphisms PSL(3,5) or PSL(4,3) using the method of E. S. Kramer and D. M. Mesner [Discrete Math. 15, 263–296 (1976; Zbl 0362.05049)].

In this article, we investigate the existence of large sets of 3-designs of prime sizes with prescribed groups of automorphisms PSL(2,q) and PGL(2,q) for q < 60. We also construct some new interesting large sets by the use of the computer program DISCRETA. The results obtained through these direct methods along with known recursive constructions ar...

The purpose of this study was to present an equation that accurately predicts 1 repetition maximum (RM) over a wide range of repetitions to fatigue (RTF) for 4 different machine resistance exercises in postmenopausal women. Seventy trained women (age = 57.4 +/- 3.1 years) performed maximal and submaximal repetitions on leg press, bench press, rowin...

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...

New linear codes (sometimes optimal) over the finite field with q elements are constructed. In order to do this, an equivalence between the existence of a linear code with a prescribed minimum distance and the existence of a solution of a certain system of Diophantine linear equations is used. To reduce the size of the system of equations, the sear...

The minimal cardinality of a q-ary code of length n and covering radius at most R is denoted by Kq(n, R); if we have the additional requirement that the minimum distance be at least d, it is denoted by Kq(n, R, d). Obviously, Kq(n, R, d) = Kq(n, R). In this paper, we study instances for which Kq(n,1,2) > Kq(n, 1) and, in particular, determine K4(4,...

This correspondence revisits the idea of constructing a binary [mn,mk] code from an [n,k] code over F₂^m by concatenating the code with a suitable basis representation of F₂^m over F₂. We construct two nonequivalent examples of doubly even self-dual binary codes of length 160 which turn out to be...

In this correspondence, we compute the weight enumerators of various quadratic residue codes over F₂ and F₃, together with certain codes of related families like the duadic and the quadratic double circulant codes. We use a parallel algorithm to find the number of codewords of a given (not too high) weight, from which we deduc...

We compute the weight enumerators of various quadratic residue (QR) codes over F2 and F3, together with certain codes of related families like the duadic codes. We use a parallel algorithm to find the number of codewords of a given (not too high) weight, from which we deduce by usual classical methods for selfdual and isodual codes over F2 and F3 t...

For the purpose of error correcting linear codes over a finite field GF (q) and fixed dimension k we are interested in codes with high minimum distance d as these allow the correction of up to b(d-1)/2c errors. On the other hand we are in-terested in codes with minimum redundancy, i. e. codes of small length n. High minimum distance and small lengt...

This paper discusses the construction of binary linear Golay [160, 80] (mn,mk) codes from an extended Reed Solomon [32, 16] (n, k) code over F₃₂ (F₂^m) by concatenating the given code with a suitable basis symmetric representation of Frobenius automorphism F₃₂ over F₂ with a minimum distance 24.

Short descriptions of computer algebra systems are presented in three sections: major systems, special purpose systems, and packages. However, the separation between special purpose systems and packages is not to be taken too literally. An older survey is the paper by Calmet and van Hulzen in [Buchberger et al. 1982]. There is now an excellent new...

An attempt was made to construct binary [mn,mk] codes from an [n,k] code over F2m by concatenating the given code with a suitable basis representation of F2m over F2. It was found that regardless of the chosen symmetric representation, the resulting binary [160,80] code has a minimum distance 24, and that the resulting code is self-dual and doubly...

In the football pool problem one wants to minimize the cardinality of a ternary code, C⊆F3n, with covering radius one, and the size of a minimum code is denoted by σn. The smallest unsettled case is 63⩽σ6⩽73. The lower bound is here improved to 65 in a coordinate-by-coordinate backtrack search using the LLL algorithm and complete equivalence checki...

The market split problem was proposed by Cornuéjols and Dawande as benchmark problem for algorithms solving linear systems with 0/1 variables. Here, we present an algorithm for the more general problem A · x = b with arbitrary lower and upper bound on the variables. The algorithm consists of exhaustive enumeration of all points of a suitable lattic...

There are 107 non-isomorphic coverings of the Aztec diamond of order 5 by the 25 one-sided tetrasticks. In [6] Donald Knuth describes how to attack certain types of puzzles by computer. One type of puzzle he tried to solve is the covering of patterns by tetrasticks. Tetrasticks | or more general polysticks | were introduced by Brian Barwell [1]: Pi...