Orthogonal Latin Squares Of Order Research Articles

Mutually Orthogonal Latin Squares as Group Transversals

Abstract In this paper, we give a method to determine a complete set of mutually orthogonal Latin squares of order m, where m is an odd prime or power of a prime, as a group transversal of a Frobenius group.

Pairs of MOLS of order ten satisfying non-trivial relations

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.

Thirty-six Entangled Officers of Euler: Quantum Solution to a Classically Impossible Problem.

The negative solution to the famous problem of 36 officers of Euler implies that there are no two orthogonal Latin squares of order six. We show that the problem has a solution, provided the officers are entangled, and construct orthogonal quantum Latin squares of this size. As a consequence, we find an example of the long-elusive Absolutely Maximally Entangled state AME(4,6) of four subsystems with six levels each, equivalently a 2-unitary matrix of size 36, which maximizes the entangling power among all bipartite unitary gates of this dimension, or a perfect tensor with four indices, each running from one to six. This special state deserves the appellation golden AME state, as the golden ratio appears prominently in its elements. This result allows us to construct a pure nonadditive quhex quantum error detection code ((3,6,2))_{6}, which saturates the Singleton bound and allows one to encode a six-level state into a triplet of such states.

Recent results on Choi's orthogonal Latin squares

Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler although this was less known. He introduced a pair of orthogonal Latin squares of order 9 in his book. Interestingly, his two orthogonal non-double-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi's Latin squares of order 9. The most recent one is Ko-Wei Lih's construction of Choi's Latin squares of order 9 based on the two $3 \times 3$ orthogonal Latin squares. In this paper, we give a new generalization of Choi's orthogonal Latin squares of order 9 to orthogonal Latin squares of size $n^2$ using the Kronecker product including Lih's construction. We find a geometric description of Choi's orthogonal Latin squares of order 9 using the dihedral group $D_8$. We also give a new way to construct magic squares from two orthogonal non-double-diagonal Latin squares, which explains why Choi's Latin squares produce a magic square of order 9.

The geometric reason for the non-existence of a MOL(6)

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.

A lower bound on HMOLS with equal sized holes

It is known that N(n), the maximum number of mutually orthogonal latin squares of order n, satisfies the lower bound N(n)≥n1/14.8 for large n. For h≥2, relatively little is known about the quantity N(hn), which denotes the maximum number of ‘HMOLS’ or mutually orthogonal latin squares having a common equipartition into n holes of a fixed size h. We generalize a difference matrix method that had been used previously for explicit constructions of HMOLS. An estimate of R.M. Wilson on higher cyclotomic numbers guarantees our construction succeeds in suitably large finite fields. Feeding this into a generalized product construction, we are able to establish the lower bound N(hn)≥(log⁡n)1/δ for any δ>2 and all n>n0(h,δ).

Packing of permutations into Latin squares

For every positive integer n greater than 4 there is a set of Latin squares of order n such that every permutation of the numbers 1,…,n appears exactly once as a row, a column, a reverse row or a reverse column of one of the given Latin squares. If n is greater than 4 and not of the form p or 2p for some prime number p congruent to 3 modulo 4, then there always exists a Latin square of order n in which the rows, columns, reverse rows and reverse columns are all distinct permutations of 1,…,n, and which constitute a permutation group of order 4n. If n is prime congruent to 1 modulo 4, then a set of (n−1)∕4 mutually orthogonal Latin squares of order n can also be constructed by a classical method of linear algebra in such a way, that the rows, columns, reverse rows and reverse columns are all distinct and constitute a permutation group of order n(n−1).

Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of $t$ mutually orthogonal Latin squares of order $n$ to construct a set of $2t$ mutually orthogonal Latin squares of order $n^t$.

Pseudo orthogonal Latin squares

Abstract Two Latin squares A, B of order n are called pseudo orthogonal if for any 1 ≤ i, j ≤ n there exists a k, 1 ≤ k ≤ n, such that A(i, k) = B(j, k). We prove that the existence of a family of m mutually pseudo orthogonal Latin squares of order n is equivalent to the existence of a family of m mutually orthogonal Latin squares of order n. We also obtain exact values of clique partition numbers of several classes of complete multipartite graphs and of the tensor product of complete graphs.

Uniform semi-Latin squares and their pairwise-variance aberrations

For integers n>2 and k>0, an (n×n)∕ksemi-Latin square is an n×n array of k-subsets (called blocks) of an nk-set (of treatments), such that each treatment occurs once in each row and once in each column of the array. A semi-Latin square is uniform if every pair of blocks, not in the same row or column, intersect in the same positive number of treatments. It is known that a uniform (n×n)∕k semi-Latin square is Schur optimal in the class of all (n×n)∕k semi-Latin squares, and here we show that when a uniform (n×n)∕k semi-Latin square exists, the Schur optimal (n×n)∕k semi-Latin squares are precisely the uniform ones. We then compare uniform semi-Latin squares using the criterion of pairwise-variance (PV) aberration, introduced by J. P. Morgan for affine resolvable designs, and determine the uniform (n×n)∕k semi-Latin squares with minimum PV aberration when there exist n−1 mutually orthogonal Latin squares of order n. These do not exist when n=6, and the smallest uniform semi-Latin squares in this case have size (6×6)∕10. We present a complete classification of the uniform (6×6)∕10 semi-Latin squares, and display the one with least PV aberration. We give a construction producing a uniform ((n+1)×(n+1))∕((n−2)n) semi-Latin square when there exist n−1 mutually orthogonal Latin squares of order n, and determine the PV aberration of such a uniform semi-Latin square. Finally, we describe how certain affine resolvable designs and balanced incomplete-block designs can be constructed from uniform semi-Latin squares. From the uniform (6×6)∕10 semi-Latin squares we classified, we obtain (up to block design isomorphism) exactly 16875 affine resolvable designs for 72 treatments in 36 blocks of size 12 and 8615 balanced incomplete-block designs for 36 treatments in 84 blocks of size 6. In particular, this shows that there are at least 16875 pairwise non-isomorphic orthogonal arrays OA(72,6,6,2).

Construction of Mutually Unbiased Bases Using Mutually Orthogonal Latin Squares

Using character of mutually orthogonal Latin squares, we first prove that two special squares are orthogonal to a complete set of mutually orthogonal Latin squares of order d. Then using the complete set of mutually orthogonal Latin squares of order d and two special squares, we construct d + 1 mutually unbiased bases in $\mathbb {C}^{d} \otimes \mathbb {C}^{d}$, which include d − 1 mutually unbiased maximally entangled bases and two mutually unbiased product bases. We also present the corresponding construction in $\mathbb {C}^{3} \otimes \mathbb {C}^{3}$, $\mathbb {C}^{4} \otimes \mathbb {C}^{4}$ and $\mathbb {C}^{5} \otimes \mathbb {C}^{5}$.

Triples of orthogonal Latin and Youden rectangles for small orders

AbstractWe have performed a complete enumeration of nonisotopic triples of mutually orthogonal Latin rectangles for . Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this, we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of rectangles which are formed by extending mutually orthogonal triples with nontrivial autotopisms one row at a time, and requiring that the autotopism group is nontrivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of rectangles, formed by juxtaposing two selected copies of complete sets of mutually orthogonal Latin squares of order 4.

On the number of SQSs, latin hypercubes and MDS codes

AbstractWe 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 .

Применение высокопроизводительных вычислений для поиска троек взаимно частично ортогональных диагональных латинских квадратов порядка 10

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.

Mutually orthogonal latin squares with large holes

Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once. This definition extends naturally to ‘incomplete’ latin squares each having a hole on the same rows, columns, and symbols. If an incomplete latin square of order n has a hole of order m, then it is an easy observation that n≥2m. More generally, if a set of t incomplete mutually orthogonal latin squares of order n have a common hole of order m, then n≥(t+1)m. In this article, we prove such sets of incomplete squares exist for all n,m≫0 satisfying n≥8(t+1)2m.

A polynomial embedding of pairs of orthogonal partial Latin squares

We show that a pair of orthogonal partial Latin squares of order n can be embedded in a pair of orthogonal Latin squares of order at most 16n4 and all orders greater than or equal to 48n4. This paper provides the first direct polynomial order embedding construction for pairs of orthogonal partial Latin squares.

On the number of transversal designs

Bounds are obtained on the number of distinct transversal designs TD(g,n) (having g groups with n points in each group) for certain values of g and n. Amongst other results it is proved that, if 2<g⩽q+1 where q is a prime power, then the number of nonisomorphic TD(g,qr) designs is at least qαrq2r(1−o(1)) as r→∞, where α=1/q4. The bounds obtained give equivalent bounds for the numbers of distinct and nonisomorphic sets of g−2 mutually orthogonal Latin squares of order n in the corresponding cases. Applications to other combinatorial designs are also described.

Construction of bimagic squares using orthogonal Sudoku squares (extended abstract)

Abstract It is well-known that Tarry was the first to prove that orthogonal latin squares of order six do not exist. Less well-known is that he was the first to give constructions for bimagic squares valid (in theory) for all orders p 2 , where p is a prime. He used in effect pairs of orthogonal diagonal Sudoku squares, the ones used for a particular prime p being determined by an appropriate “key”. [A square is bimagic if it is a magic square and remains magic when all its entries are replaced by their squares.] We show that one of the pairs of orthogonal diagonal Sudoku squares which is appropriate for the construction when p = 3 can be generalized to provide a standard construction valid for all primes p except p = 5 . We explain why the construction fails when p = 5 .

On the existence of three dimensional Room frames and Howell cubes

A Howell design of side s and order 2n+2, or more briefly an H(s,2n+2), is an s×s array in which each cell is either empty or contains an unordered pair of elements from some 2n+2 set V such that (1) every element of V occurs in precisely one cell of each row and each column, and (2) every unordered pair of elements from V is in at most one cell of the array. It follows immediately from the definition of an H(s,2n+2) that n+1≤s≤2n+1. A d-dimensional Howell design Hd(s,2n+2) is a d-dimensional array of side s such that (1) every cell is either empty or contains an unordered pair of elements from some 2n+2 set V, and (2) each two-dimensional projection is an H(s,2n+2). The two boundary cases are well known designs: an Hd(2n+1,2n+2) is a Room d-cube of side 2n+1 and the existence of d mutually orthogonal latin squares of order n+1 implies the existence of an Hd(n+1,2n+2). In this paper, we investigate the existence of Howell cubes, H3(s,2n+2). We completely determine the spectrum for H3(2n,2n+α) where α∈{2,4,6,8}. In addition, we establish the existence of 3-dimensional Room frames of type 2v for all v≥5 with only a few small possible exceptions for v.

Four Mutually Orthogonal Latin Squares of Order 14

AbstractThe paper gives example of orthogonal array OA(6, 14) obtained from a difference matrix . The construction is equivalent to four mutually orthogonal Latin squares (MOLS) of order 14. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 363–367, 2012