arXiv subjects

Patric R. J. Östergård

Publications and source records attributed to Patric R. J. Östergård.

At least 19 recordsLinked to original sources

arXiv2026

Equivalence of Complex Hadamard Matrices

Symmetry in the context of equivalence or isomorphism is a fundamental and natural concept in any study of discrete structures. Symmetries are also important for non-discrete structures, but their treatment can be more challenging and is perhaps therefore often overlooked. This holds for many studies of complex Hadamard matrices, that is, matrices with unimodular complex entries satisfying the equation $HH^{\dagger} = nI$, where $H^{\dagger}$ is the conjugate transpose of $H$. In the current work, equivalence of complex Hadamard matrices is considered, and algorithms for determining equivalence of matrices and the automorphism group of a matrix are presented. The algorithms are used to establish the automorphism group of a large number of complex Hadamard matrices from the literature.

math.CO↗

arXiv2023

Enumerating Steiner Triple Systems

Steiner triple systems (STSs) have been classified up to order 19. Earlier estimations of the number of isomorphism classes of STSs of order 21, the smallest open case, are discouraging as for classification, so it is natural to focus on the easier problem of merely counting the isomorphism classes. Computational approaches for counting STSs are here considered and lead to an algorithm that is used to obtain the number of isomorphism classes for order 21: 14,796,207,517,873,771.

math.CO↗

arXiv2023

Constructing Random Steiner Triple Systems: An Experimental Study

Several methods for generating random Steiner triple systems (STSs) have been proposed in the literature, such as Stinson's hill-climbing algorithm and Cameron's algorithm, but these are not yet completely understood. Those algorithms, as well as some variants, are here assessed for STSs of both small and large orders. For large orders, the number of occurrences of certain configurations in the constructed STSs are compared with the corresponding expected values of random hypergraphs. Modifications of the algorithms are proposed.

math.CO↗

arXiv2022

Steiner Triple Systems of Order 21 with Subsystems

The smallest open case for classifying Steiner triple systems is order 21. A Steiner triple system of order 21, an STS(21), can have subsystems of orders 7 and 9, and it is known that there are 12,661,527,336 isomorphism classes of STS(21)s with sub-STS(9)s. Here, the classification of STS(21)s with subsystems is completed by settling the case of STS(21)s with sub-STS(7)s. There are 116,635,963,205,551 isomorphism classes of such systems. An estimation of the number of isomorphism classes of STS(21)s is given.

math.CO↗

arXiv2021

Algorithms and Complexity for Counting Configurations in Steiner Triple Systems

Steiner triple systems form one of the most studied classes of combinatorial designs. Configurations, including subsystems, play a central role in the investigation of Steiner triple systems. With sporadic instances of small systems, ad-hoc algorithms for counting or listing configurations are typically fast enough for practical needs, but with many systems or large systems, the relevance of computational complexity and algorithms of low complexity is highlighted. General theoretical results as well as specific practical algorithms for important configurations are presented.

math.CO↗

arXiv2021

Switching 3-edge-colorings of cubic graphs

The chromatic index of a cubic graph is either 3 or 4. Edge-Kempe switching, which can be used to transform edge-colorings, is here considered for 3-edge-colorings of cubic graphs. Computational results for edge-Kempe switching of cubic graphs up to order 30 and bipartite cubic graphs up to order 36 are tabulated. Families of cubic graphs of orders $4n+2$ and $4n+4$ with $2^n$ edge-Kempe equivalence classes are presented; it is conjectured that there are no cubic graphs with more edge-Kempe equivalence classes. New families of nonplanar bipartite cubic graphs with exactly one edge-Kempe equivalence class are also obtained. Edge-Kempe switching is further connected to cycle switching of Steiner triple systems, for which an improvement of the established classification algorithm is presented.

math.CO↗

arXiv2019

On the Structure of Small Strength-$2$ Covering Arrays

A covering array $\rm{CA}(N;t,k,v)$ of strength $t$ is an $N \times k$ array of symbols from an alphabet of size $v$ such that in every $N \times t$ subarray, every $t$-tuple occurs in at least one row. A covering array is \emph{optimal} if it has the smallest possible $N$ for given $t$, $k$, and $v$, and \emph{uniform} if every symbol occurs $\lfloor N/v \rfloor$ or $\lceil N/v \rceil$ times in every column. Prior to this paper the only known optimal covering arrays for $t=2$ were orthogonal arrays, covering arrays with $v=2$ constructed from Sperner's Theorem and the Erdős-Ko-Rado Theorem, and eleven other parameter sets with $v>2$ and $N > v^2$. In all these cases, there is a uniform covering array with the optimal size. It has been conjectured that there exists a uniform covering array of optimal size for all parameters. In this paper a new lower bound as well as structural constraints for small uniform strength-$2$ covering arrays are given. Moreover, covering arrays with small parameters are studied computationally. The size of an optimal strength-$2$ covering array with $v > 2$ and $N > v^2$ is now known for $21$ parameter sets. Our constructive results continue to support the conjecture.

math.CO↗

arXiv2018

Constructions of maximum few-distance sets in Euclidean spaces

A finite set of distinct vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called an $s$-distance set if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality $s$. In this paper we present a combined approach of isomorph-free exhaustive generation of graphs and Gröbner basis computation to classify the largest $3$-distance sets in $\mathbb{R}^4$, the largest $4$-distance sets in $\mathbb{R}^3$, and the largest $6$-distance sets in $\mathbb{R}^2$. We also construct new examples of large $s$-distance sets for $d\leq 8$ and $s\leq 6$, and independently verify several earlier results from the literature.

math.MG↗

arXiv2017

LCL problems on grids

LCLs or locally checkable labelling problems (e.g. maximal independent set, maximal matching, and vertex colouring) in the LOCAL model of computation are very well-understood in cycles (toroidal 1-dimensional grids): every problem has a complexity of $O(1)$, $Θ(\log^* n)$, or $Θ(n)$, and the design of optimal algorithms can be fully automated. This work develops the complexity theory of LCL problems for toroidal 2-dimensional grids. The complexity classes are the same as in the 1-dimensional case: $O(1)$, $Θ(\log^* n)$, and $Θ(n)$. However, given an LCL problem it is undecidable whether its complexity is $Θ(\log^* n)$ or $Θ(n)$ in 2-dimensional grids. Nevertheless, if we correctly guess that the complexity of a problem is $Θ(\log^* n)$, we can completely automate the design of optimal algorithms. For any problem we can find an algorithm that is of a normal form $A' \circ S_k$, where $A'$ is a finite function, $S_k$ is an algorithm for finding a maximal independent set in $k$th power of the grid, and $k$ is a constant. Finally, partially with the help of automated design tools, we classify the complexity of several concrete LCL problems related to colourings and orientations.

cs.DC↗

arXiv2017

Enumeration of Seidel matrices

In this paper Seidel matrices are studied, and their spectrum and several related algebraic properties are determined for order $n\leq 13$. Based on this Seidel matrices with exactly three distinct eigenvalues of order $n\leq 23$ are classified. One consequence of the computational results is that the maximum number of equiangular lines in $\mathbb{R}^{12}$ with common angle $1/5$ is exactly $20$.

math.CO↗

arXiv2016

Constructing error-correcting binary codes using transitive permutation groups

Let $A_2(n,d)$ be the maximum size of a binary code of length $n$ and minimum distance $d$. In this paper we present the following new lower bounds: $A_2(18,4) \ge 5632$, $A_2(21,4) \ge 40960$, $A_2(22,4) \ge 81920$, $A_2(23,4) \ge 163840$, $A_2(24,4) \ge 327680$, $A_2(24,10) \ge 136$, and $A_2(25,6) \ge 17920$. The new lower bounds are a result of a systematic computer search over transitive permutation groups.

cs.IT↗

arXiv2016

There is No McLaughlin Geometry

We determine that there is no partial geometry ${\cal G}$ with parameters $(s,t,α)=(4,27,2)$. The existence of such a geometry has been a challenging open problem of interest to researchers for almost 40 years. The particular interest in ${\cal G}$ is due to the fact that it would have the exceptional McLaughlin graph as its point graph. Our proof makes extensive use of symmetry and high-performance distributed computing, and details of our techniques and checks are provided. One outcome of our work is to show that a pseudogeometric strongly regular graph achieving equality in the Krein bound need not be the point graph of any partial geometry.

math.CO↗

arXiv2016

The chromatic number of the square of the 8-cube

A cube-like graph is a Cayley graph for the elementary abelian group of order $2^n$. In studies of the chromatic number of cube-like graphs, the $k$th power of the $n$-dimensional hypercube, $Q_n^k$, is frequently considered. This coloring problem can be considered in the framework of coding theory, as the graph $Q_n^k$ can be constructed with one vertex for each binary word of length $n$ and edges between vertices exactly when the Hamming distance between the corresponding words is at most $k$. Consequently, a proper coloring of $Q_n^k$ corresponds to a partition of the $n$-dimensional binary Hamming space into codes with minimum distance at least $k+1$. The smallest open case, the chromatic number of $Q_8^2$, is here settled by finding a 13-coloring. Such 13-colorings with specific symmetries are further classified.

math.CO↗

arXiv2015

Planar Hypohamiltonian Graphs on 40 Vertices

A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer-aided generation of certain families of planar graphs with girth 4 and a fixed number of 4-faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles are replaced by Hamiltonian paths throughout the definition of hypohamiltonian graphs, we get the definition of hypotraceable graphs. It is shown that there is a planar hypotraceable graph of order 154 and of all orders greater than or equal to 156. We also show that the smallest hypohamiltonian planar graph of girth 5 has 45 vertices.

math.CO↗

arXiv2015

A coloring of the square of the 8-cube with 13 colors

Let $χ_{\bar{k}}(n)$ be the number of colors required to color the $n$-dimensional hypercube such that no two vertices with the same color are at a distance at most $k$. In other words, $χ_{\bar{k}}(n)$ is the minimum number of binary codes with minimum distance at least $k+1$ required to partition the $n$-dimensional Hamming space. By giving an explicit coloring, it is shown that $χ_{\bar{2}}(8)=13$.

math.CO↗

arXiv2015

Further Results on the Classification of MDS Codes

A $q$-ary maximum distance separable (MDS) code $C$ with length $n$, dimension $k$ over an alphabet $\mathcal{A}$ of size $q$ is a set of $q^k$ codewords that are elements of $\mathcal{A}^n$, such that the Hamming distance between two distinct codewords in $C$ is at least $n-k+1$. Sets of mutually orthogonal Latin squares of orders $q\leq 9$, corresponding to two-dimensional \mbox{$q$-}ary MDS codes, and $q$-ary one-error-correcting MDS codes for $q\leq 8$ have been classified in earlier studies. These results are used here to complete the classification of all $7$-ary and $8$-ary MDS codes with $d\geq 3$ using a computer search.

math.CO↗

arXiv2015

New Lower Bounds for the Shannon Capacity of Odd Cycles

The Shannon capacity of a graph $G$ is defined as $c(G)=\sup_{d\geq 1}(α(G^d))^{\frac{1}{d}},$ where $α(G)$ is the independence number of $G$. The Shannon capacity of the cycle $C_5$ on $5$ vertices was determined by Lovász in 1979, but the Shannon capacity of a cycle $C_p$ for general odd $p$ remains one of the most notorious open problems in information theory. By prescribing stabilizers for the independent sets in $C_p^d$ and using stochastic search methods, we show that $α(C_7^5)\geq 350$, $α(C_{11}^4)\geq 748$, $α(C_{13}^4)\geq 1534$ and $α(C_{15}^3)\geq 381$. This leads to improved lower bounds on the Shannon capacity of $C_7$ and $C_{15}$: $c(C_7)\geq 350^{\frac{1}{5}}> 3.2271$ and $c(C_{15})\geq 381^{\frac{1}{3}}> 7.2495$.

cs.IT↗

arXiv2014

On the Classification of MDS Codes

A $q$-ary code of length $n$, size $M$, and minimum distance $d$ is called an $(n,M,d)_q$ code. An $(n,q^{k},n-k+1)_q$ code is called a maximum distance separable (MDS) code. In this work, some MDS codes over small alphabets are classified. It is shown that every $(k+d-1,q^k,d)_q$ code with $k\geq 3$, $d \geq 3$, $q \in \{5,7\}$ is equivalent to a linear code with the same parameters. This implies that the $(6,5^4,3)_5$ code and the $(n,7^{n-2},3)_7$ MDS codes for $n\in\{6,7,8\}$ are unique. The classification of one-error-correcting $8$-ary MDS codes is also finished; there are $14$, $8$, $4$, and $4$ equivalence classes of $(n,8^{n-2},3)_8$ codes for $n=6,7,8,9$, respectively. One of the equivalence classes of perfect $(9,8^7,3)_8$ codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction.

cs.IT↗