Charles Laywine

Discrete mathematics using Latin squares

LATIN SQUARES. A Brief Introduction to Latin Squares. Mutually Orthogonal Latin Squares. GENERALIZATIONS. Orthogonal Hypercubes. Frequency Squares. RELATED MATHEMATICS. Principle of Inclusion--Exclusion. Groups and Latin Squares. Graphs and Latin Squares. APPLICATIONS. Affine and Projective Planes. Orthogonal Hypercubes and Affine Designs. Magic Squares. Room Squares. Statistics. Error--Correcting Codes. Cryptology. (t,m,s)--Nets. Miscellaneous Applications of Latin Squares. Appendices. Indexes.

D-Dimensional hypercubes and the Euler and MacNeish conjectures

We give a construction for large sets of mutually orthogonal hypercubes of dimensionald given sets of mutually orthogonal latin squares and hypercubes of lower dimension. We also considerd>-2 dimensional versions of the Euler and MacNeish conjectures as well as discussing applications to improved constructions of (t, m, s)-nets, useful in pseudorandom number generation and quasi-Monte-Carlo methods of numerical integration.

Generalizations of Bose's equivalence between complete sets of mutually orthogonal Latin squares and affine planes

Abstract Using affine resolvable designs and complete sets of mutually orthogonal frequency squares and hypercubes, we provide several generalizations of Boses equivalence between affine planes of order n and complete sets of mutually orthogonal latin squares of order n. We also characterize those complete sets of orthogonal frequency squares and hypercubes which are equivalent to affine geometries.

A simple construction giving the two non-isomorphic triangle-free 3-colored K 16 's

We present a simple construction which exhibits the two non-isomorphic triangle-free 3-colored K16s. The construction enables us to clarify the relation between these two graphs.

An expression for the number of equivalence classes of Latin squares under row and column permutations

Abstract In this paper equivalence classes of Latin squares induced by row and column permutations are considered. An expression for the number of such classes for an n th order Latin square is obtained in terms of Latin rectangles with n rows. In the case where n is prime the expression gives a simple result for those squares invariant under a permutation other than the identity.

Subsquares in orthogonal Latin squares as subspaces in affine geometries: a generalization of an equivalence of Bose

The combinatorial properties of subsquares in orthogonal latin squares are examined. Using these properties it is shown that in appropriate orthogonal latin squares of ordermh blocks of subsquares of ordermh(i−1)/i, wherei dividesh, form the hyperplanes of the affine geometryAG (2i, mh/i). This means that a given set of mutually orthogonal latin squares may be equivalent simultaneously to a number of different geometries depending on the order of the subsquares used to form the hyperplanes. In the case thati=1, the subsquares become points, the hyperplanes become lines, and the equivalence reduces to the well known result of Bose relating orthogonal latin squares and affine planes.

An affine design with v = m2h and k = m2h?1 not equivalent to a complete set of F(mh; mh?1) MOFS

Existing sufficient conditions for the construction of a complete set of mutually orthogonal frequency squares from an affine resolvable design are improved to give necessary and sufficient conditions. In doing so a design is exhibited that proves that the class of complete sets of MOFS under consideration is a proper subset of the class of affine resolvable designs with matching parameters.

A counter-example to a conjecture relating complete sets of frequency squares and affine geometries

Abstract By generalizing the construction of complete sets of mutually orthogonal latin squares from affine planes, Mullen (1988) showed how to obtain complete sets of mutually orthogonal frequency squares from affine geometries. In this paper, the construction of a complete set of frequency squares not equivalent to an affine geometry is outlined. This set eliminates any hope of finding a straightforward generalization of Boses equivalence between latin squares and affine planes by means of frequency squares and affine geometries.

A derivation of an affine plane of order 4 from a triangle-free 3-colored K16

Abstract A variety of results connecting latin squares and graphs of different types, are known. In this paper a new relationship is given through the derivation of AG(2,4), the affine plane of order 4, from the 3-colored, triangle-free K16 constructed by Greenwood and Gleason in the proof that the classic Ramsey number R(3,3,3)=17. In the derivation each line of this affine plane is defined by a set of 4 vertices of the K16, which are mutually connected by edges of three colors so that each color defines one of three 1-factor of that K4.