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12
Small latin squares, quasigroups and loops
 Journal of Combinatorial Designs
, 2007
"... We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 (Kolesova, Lam and Thiel, 1990), quasigroups of order 6 (Bower, 2000) and loops of ..."
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Cited by 11 (4 self)
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We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 (Kolesova, Lam and Thiel, 1990), quasigroups of order 6 (Bower, 2000) and loops of order 7 (Brant and Mullen, 1985). The loops of order 8 have been independently found by \QSCGZ" and Guerin (unpublished, 2001). We also report on the most extensive search so far for a triple of mutually orthogonal Latin squares (MOLS) of order 10. Our computations show that any such triple must have only squares with trivial symmetry groups. 1
An Orderly Algorithm and Some Applications in Finite Geometry
 Discrete Math
, 1996
"... An algorithm for generating combinatorial structures is said to be an orderly algorithm if it produces precisely one representative of each isomorphism class. In this paper we describe a way to construct an orderly algorithm that is suitable for several common searching tasks in combinatorics. We ..."
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Cited by 11 (2 self)
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An algorithm for generating combinatorial structures is said to be an orderly algorithm if it produces precisely one representative of each isomorphism class. In this paper we describe a way to construct an orderly algorithm that is suitable for several common searching tasks in combinatorics. We illustrate this with examples of searches in finite geometry, and an extended application where we classify all the maximal partial flocks of the hyperbolic and elliptic quadrics in PG(3; q) for q 13.
There are 526,915,620 nonisomorphic onefactorizations of K 12
 J. COMBIN. DES
, 1994
"... We enumerate the nonisomorphic and the distinct onefactorizations of K 12 . We also describe the algorithm used to obtain the result, and the methods we used to verify these numbers. ..."
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Cited by 9 (1 self)
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We enumerate the nonisomorphic and the distinct onefactorizations of K 12 . We also describe the algorithm used to obtain the result, and the methods we used to verify these numbers.
Steiner Triple Systems of Order 19 with Nontrivial Automorphism Group
 Math. Comp
, 2003
"... There are 172,248 Steiner triple systems of order 19 having a nontrivial automorphism group. Computational methods suitable for generating these designs are developed. ..."
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Cited by 7 (1 self)
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There are 172,248 Steiner triple systems of order 19 having a nontrivial automorphism group. Computational methods suitable for generating these designs are developed.
Efficient Exhaustive Listings of Reversible One Dimensional Cellular Automata
"... Algebra From a rectangular structure R, using the bijection d from equation (55) above and denoting by R(s; t) the unique rectangle on the pair (s; t) guaranteed by (52), define ffl : S \Theta S ! S (63) (s; t) 7! u where fug = (d \Gamma1 (s)) 2 " (d \Gamma1 (t)) 1 ffi : S \Theta S ! S (64) ( ..."
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Cited by 5 (2 self)
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Algebra From a rectangular structure R, using the bijection d from equation (55) above and denoting by R(s; t) the unique rectangle on the pair (s; t) guaranteed by (52), define ffl : S \Theta S ! S (63) (s; t) 7! u where fug = (d \Gamma1 (s)) 2 " (d \Gamma1 (t)) 1 ffi : S \Theta S ! S (64) (s; t) 7! u where fug = d(R(s; t)) as binary operations on S.
Comparism of radius 1/2 and radius 1 Paradigms in One Dimensional Reversible Cellular Automata
"... We compare the usefulness of the paradigms of radius 1 and radius 1=2 reversible cellular automata. Although the two techniques are equivalent, it is seen that the radius 1=2 paradigm leads to clearer theory, a combinatorial model with good explicit constructions and possibly to a better intuition. ..."
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Cited by 4 (0 self)
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We compare the usefulness of the paradigms of radius 1 and radius 1=2 reversible cellular automata. Although the two techniques are equivalent, it is seen that the radius 1=2 paradigm leads to clearer theory, a combinatorial model with good explicit constructions and possibly to a better intuition. The results include related construction methods for both paradigms, in the radius 1=2 case an exhaustive enumeration is possible. 1 Introduction Cellular Automata theory is, in general, plagued by varying definitions and assumptions, overlaid with a network of equivalences between various points of view, paradigms and models. The current paper addresses two differing but equivalent points of view of cellular automata, concentrating upon the case of reversible cellular automata. The main body of results in Section 3 come from the author's thesis [3], the results in section 4 are an attempt to use similar tools. The way in which these tools are unusable in the radius 1 paradigm used in Secti...
Discrete Mathematics for Combinatorial Chemistry
, 1998
"... The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry. ..."
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Cited by 2 (1 self)
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The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry.
Search and Enumeration Techniques for Incidence Structures
, 1998
"... This thesis investigates a number of probabilistic and exhaustive computational search techniques for the construction of a wide variety of combinatorial designs, and in particular, incidence structures. The emphasis is primarily from a computer science perspective, and focuses on the algorithmic de ..."
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Cited by 2 (0 self)
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This thesis investigates a number of probabilistic and exhaustive computational search techniques for the construction of a wide variety of combinatorial designs, and in particular, incidence structures. The emphasis is primarily from a computer science perspective, and focuses on the algorithmic development of the techniques, taking into account running time considerations and storage requirements. The search and enumeration techniques developed in this thesis have led to the discovery of a number of new results in the field of combinatorial design theory. Page ii Page iii Acknowledgments I would like to extend my sincere thanks to a number of people who have given me a great deal of assistance and support throughout the preparation of this thesis. Firstly, my supervisor Peter Gibbons. I am very grateful for the encouragement and guidance he has given to me. His remarkable enthusiasm and friendliness have helped to make this thesis a most enjoyable experience. My family, for their...
Computers and Discovery in Algebraic Graph Theory
 Edinburgh, 2001), Linear Algebra Appl
, 2001
"... We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. ..."
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Cited by 1 (0 self)
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We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.
Algorithms for Group Actions: Homomorphism Principle and Orderly Generation Applied to Graphs
 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1996
"... The generation of discrete structures up to isomorphism is interesting as well for theoretical as for practical purposes. Mathematicians want to look at and analyse structures and for example chemical industry uses mathematical generators of isomers for structure elucidation. The example chosen in t ..."
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Cited by 1 (1 self)
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The generation of discrete structures up to isomorphism is interesting as well for theoretical as for practical purposes. Mathematicians want to look at and analyse structures and for example chemical industry uses mathematical generators of isomers for structure elucidation. The example chosen in this paper for explaining general generation methods is a relatively far reaching and fast graph generator which should serve as a basis for the next more powerful version of MOLGEN, our generator of chemical isomers. 1