On Methods of Constructing Sets of Mutually Orthogonal Latin Squares Using a Computer. I

This paper deals with the search for triples of partially orthogonal Latin squares of order 10. For every known pair of orthogonal Latin squares of order 10 we add a third diagonal Latin square in such a way that the orthogonality condition between it and squares from a considered pair coincides in the maximum possible number of cells. Two approaches are used: the first one is based on reducing an original problem to Boolean satisfiability problem; the second one is based on Brute force method. Several triples of the aforementioned kind with high characteristics were constructed. The experiments were held in the volunteer computing project SAT@home and on a computing cluster.

Maximal partial spreads in PG(3,4) and maximal sets of mutually orthogonal latin squares of order 16

The maximum number of mutually orthogonal Sudoku Latin squares (MOSLS) of order $$n=m^2$$n=m2 is $$n-m$$n-m. In this paper, we construct for $$n=q^2$$n=q2, q a prime power, a set of $$q^2-q-1$$q2-q-1 MOSLS of order $$q^2$$q2 that cannot be extended to a set of $$q^2-q$$q2-q MOSLS. This contrasts to the theory of ordinary Latin squares of order n, where each set of $$n-2$$n-2 mutually orthogonal Latin Squares (MOLS) can be extended to a set of $$n-1$$n-1 MOLS (which is best possible). For this proof, we construct a particular maximal partial spread of size $$q^2-q+1$$q2-q+1 in $$\mathrm {PG}(3,q)$$PG(3,q) and use a connection between Sudoku Latin squares and projective geometry, established by Bailey, Cameron and Connelly.

Chapter 11 Latin Squares and Geometry

Recently, A. B. Evans proved the following Theorem: There is a maximal set of (p − 3)/2 [(resp. (p − 1)/2] mutually orthogonal Latin squares of order p if p is a prime p ≡ 3 mod 4 (resp. p ≡ 1 mod 4). In this article I will give a slightly different proof using more geometric arguments and results of Rédei. Further, I discuss possible generalizations of Evans' Theorem to the case of Latin squares whose order is a prime power. © 1993 John Wiley & Sons, Inc.

Embedding partial Latin squares in Latin squares with many mutually orthogonal mates

We consider orthogonal arrays of strength two and even order $q$ having $n$ columns which are equivalent to $n-2$ mutually orthogonal Latin squares of order $q$. We show that such structures induce graphs on $n$ vertices, invariant up to complementation. Previous methods worked only for single Latin squares of even order and were harder to apply. If $q$ is divisible by 4, the invariant graph is simple undirected. If $q$ is 2 modulo 4, the graph is a tournament. When $n=q+1$ is maximal, the array corresponds to an affine plane, and the vertex valencies of the graph have parity $q/2$ modulo 2. We give the graphs at all possible points and lines for 22 planes of order 16. Four of the planes, none of them of translation or dual translation type, produce nonempty graphs at some points.

We establish that the logarithm of the number of latin d‐cubes of order n is and the logarithm of the number of sets of t ( is fixed) orthogonal latin squares of order n is . Similar estimations are obtained for systems of mutually strongly orthogonal latin d‐cubes. As a consequence, we construct a set of Steiner quadruple systems of order n such that the logarithm of its cardinality is as and .

Computer investigation of orthogonal Latin squares of order ten

A relation on a \(k\text {{-}net}(n)\) (or, equivalently, a set of \(k-2\) mutually orthogonal Latin squares of order n) is an \({\mathbb {F}}_{2}\) linear dependence within the incidence matrix of the net. Dukes and Howard (2014) showed that any \(6\text {{-}net}(10)\) satisfies at least two non-trivial relations, and classified the relations that could appear in such a net. We find that, up to equivalence, there are \(18\,526\,320\) pairs of MOLS satisfying at least one non-trivial relation. None of these pairs extend to a triple. We also rule out one other relation on a set of 3-MOLS from Dukes and Howard’s classification.

The problem of Euler concerning the 36 officers, (Euler, in Leonardi Euleri Opera Ser I 7:291–392, 1782), was first solved by Tarry (Comptes rendus Ass Franc Sci Nat 1 (1900) 2:170–203, 1901). Short proofs for the non-existence were given in Betten (Unterricht 36:449–453, 1983), Beth et al. (Design Theory, Bibl. Inst. Mannheim, Wien, Zürich, 1985), Stinson (J Comb Theory A 36:373–376, 1984). This problem is equivalent to the existence of a MOL(6), i. e., a pair of mutually orthogonal latin squares of order 6. Therefore in Betten (Mitt Math Ges Hamburg 39:59–76, 2019; Res Math 76:9, 2021; Algebra Geom 62:815–821, 2021) the structure of a (hypothetical) MOL(6) was studied. Now we combine the old proofs and the new studies and filter out a simple way for the proof of non-existence. The aim is not only to give still other short proofs, but to analyse the problem and reveal the geometric reason for the non-existence of a MOL(6)- and the non-solvability of Euler’s problem.

Using the tools of algebraic coding theory, we give a new proof of the nonexistence of two mutually orthogonal Latin squares of order 6.

The main purpose of this paper is to show how partial recursive functions and isols can be used to generalize the following three well-known theorems of combinatorial theory.(I) For every finite projective plane Π there is a unique number n such that Π has exactly n2 + n + 1 points and exactly n2 + n + 1 lines.(II) Every finite projective plane of order n can be coordinatized by a finite planar ternary ring of order n. Conversely, every finite planar ternary ring of order n coordinatizes a finite projective plane of order n.(III) There exists a finite projective plane of order n if and only if there exist n − 1 mutually orthogonal Latin squares of order n.

An orthogonal latin square graph (OLSG) is one in which the vertices are latin squares of the same order and on the same symbols, and two vertices are adjacent if and only if the latin squares are orthogonal. If G is an arbitrary finite graph, we say that G is realizable as an OLSG if there is an OLSG isomorphic to G. The spectrum of G [Spec(G)] is defined as the set of all integers n that there is a realization of G by latin squares of order n. The two basic theorems proved here are (1) every graph is realizable and (2) for any graph G, Spec G contains all but a finite set of integers. A number of examples are given that point to a number of wide open questions. An example of such a question is how to classify the graphs for which a given n lies in the spectrum.

Diagonally cyclic latin squares