Orthogonal Latin Squares Of Order Research Articles

Three mutually orthogonal Latin squares

In a famous paper [3], Bose, Shrikhande, and Parker proved the existence of a pair of orthogonal Latin squares of order v for all v ≠ 2,6. In the present paper it is shown that there exist three mutually orthogonal Latin squares for all v ≡ 0,1 (mod 4). This result will be needed in several future papers on the covering of pairs by quadruples.

An extension of mann's theorem to a triple of mutually orthogonal latin squares of order 10

It is shown that, given an OA(10, 5), at least one of the 5 3 contained in it has a subsquare of side 5 in which some five of the ten digits occur at least 9 and at most 12 times.

A Set of Three Mutually Orthogonal Latin Squares of Order 15

In this paper we have exhibited a set of three mutually orthogonal latin squares of order 15 associated with a Kirkman-Steiner triple system of the corresponding order.

A Set of Three Mutually Orthogonal Latin Squares of Order 15

A Set of Three Mutually Orthogonal Latin Squares of Order 15

On Embedding and Enumeration of Orthogonal Latin Squares

The importance of the theory of Latin squares and mutually orthogonal Latin squares is due to the fact that many well-known experimental designs and/or combinatorial systems are either equivalent or can be derived from Latin squares or sets of mutually orthogonal Latin squares; viz., balanced incomplete block designs, partially balanced incomplete block designs, group divisible designs, $F$-square designs, lattice designs, balanced weighing designs, orthogonal arrays, Hadamard matrices, affine planes, projective planes, partial geometries, nets, error correcting codes, strongly regular graphs and many more designs and combinatorial systems (see [7] for more details). The problem of determining whether or not a set of $r$ mutually orthogonal Latin squares of order $n$, say an $O(n, r)$ set, can be embedded in a larger set of $t$ mutually orthogonal Latin squares of order $n$, is one of the many important problems in the theory of mutually orthogonal Latin squares. An obvious and interesting problem in the area of embedding is to find a necessary and sufficient condition under which an $O(n, r)$ set can be embedded in an $O(n, t)$ set for $1 \leqq r 4$. His result does not require any knowledge about the combinatorial structure of the set $S$. Bruck [2], who generalized Shrikhande's [15] results to some extent, utilized net theory to show that any $O(n, n - 1 - d)$ set $S$ can be embedded in an $O(n, n - 1)$ set regardless of the combinatorial structure of $S$ provided that $n > (d - 1)(d^3 - d^2 + d + 2)/2$. Utilizing the above results and some group theory, some new results have been found for embedding $O(n, r)$ sets in $O(n, t)$ sets, $r < t$. The results are presented in the form of several theorems, propositions and corollaries. The first proposition is concerned with a necessary and sufficient condition for embedding $O(n, 2)$ sets in $O(n, 3)$ sets. Theorem 3.1 gives a sufficient condition for embedding an $O(n, 1)$ set in an $O(n, \lambda)$ set, $2 \leqq \lambda \leqq$ smallest prime in the prime power decomposition of $n$, less one. Theorem 3.2 states the conditions under which the method of Theorem 3.1 fails for embedding certain $O(n, 1)$ sets in a larger set. The next theorem states the conditions under which certain $O(n, 1)$ sets cannot be embedded in a larger set. Some results on properties of inverses of Latin squares and on products have also been presented. Additional results in this paper are embodied in Theorem 3.4 and relate to the enumeration of Latin squares of order $n$ for those Latin squares which have orthogonal mates and for those which do not have an orthogonal mate. Theorem 3.4 gives a lower bound for $n$ even (odd) on the number of Latin squares of degree one (at least degree two). All Latin squares of order 2, 3, 4, 5, 6 and 7 are classified with respect to degree of orthogonality.

On the Equivalence of Mann's Group Automorphism Method of Constructing an $O(n, n - 1)$ Set and Raktoe's Collineation Method of Constructing a Balanced Set of $L$-Restrictional Prime-Powered Lattice Designs

This paper demonstrates the direct relationship which exists between $O(p^m, p^m - 1)$ sets and a balanced set of $\ell$-restrictional lattice designs for $p^m$ treatments. For instance, we will show that \begin{equation*}A = \begin{bmatrix}0 & 1 1 & 1\end{bmatrix}\end{equation*} which, as has been shown by Raktoe [5] completely specifies a balanced set of 2-restrictional lattice designs for $7^2 = 49$ treatments, will also completely characterize a set of 48 mutually orthogonal latin squares of order 49, i.e. an $O(49, 48)$ set. In other words, if our interest is to exhibit an $O$(49, 48) set, the above $2 \times 2$ matrix will do the job. Strangely enough, as will be shown, $A$ also completely characterizes an $O$(4, 3) set. Note that, since a balanced set of 1-restrictional lattice designs is simply a BIB design, this paper shows in particular a different proof for the known equivalence of the $O(p^m, p^m - 1)$ sets with a class of resolvable BIB designs. Consequently, the content of this paper will be useful for those who are concerned with tabulating the designs or writing an efficient program for generating designs on a computer.

Coverings by rook domains

It is shown that a 1-dense set in V n k must contain at least n k−1/(k−1) points. As a corollary, a conjecture of Golomb and Posner on error-distributing codes is proved. It is also shown that a ( k−2)-dense set must contain at least n 2/( k−1) points. Equality can be attained if and only if k−1 divides n and there are k−2 orthogonal latin squares of order n/( k−1).

On the number of orthogonal latin squares

Let N(n) be the maximal number of mutually orthogonal Latin squares of order n and let n r be the smallest integer such that N(n)≥r for every n>n r . It is known that N(n)→∞ as n→∞ and that n 2=6. A proof is given for n 3≤51, n 5≤62 and n 29≤34, 115, 553.

The existence and construction of two families of designs for two successive experiments

SUMMARY Two families of experimental designs of the type 0: OT: TOO and 0: 00: SSS of dimensions (n + 1) x n and n x n, respectively, have been shown to exist for all n except n = 2, 3 and 6. A new method of construction for these designs has been given. These designs may be applied to situations wherein two experiments are conducted in sequence on the same experimental units and with changing treatments. In addition, a pair of orthogonal Latin squares of order ten with a common transversal has been presented and utilized to illustrate the method of construction.

On Some Methods of Construction of Partially Balanced Arrays

Partially balanced arrays are generalizations of orthogonal arrays. Multifactorial designs derived from partially balanced arrays require a reduced number of assemblies in order to accommodate a given number of factors. For instance, an orthogonal array of strength two, six symbols and four constraints, would require at least $2.6^2 = 72$ assemblies. This is because there does not exist a pair of mutually orthogonal Latin Squares of order six. But for the same situation, a partially balanced array in 42 assemblies, is constructed in this paper. The method of construction is one of composition which utilizes the existence of a pairwise partially balanced incomplete block design and an orthogonal array.

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

This paper deals with the problem of finding sets of mutually orthogonal Latin squares of order 4t (where 4t – 1 is a prime power) based on orthogonal mappings of a group. For the group G we take the module G(2, 2t) whose elements are vectors (a 1, a 2,) where a 1 is a residue class (mod 2) and a 2 is a residue class (mod 2t), the addition being defined by (a 1, a 2) + (b 1, b 2) = (c 1, c 2) where a 1 + b 1 = C 1 (mod 2) and a 2 + b 2 = c 2 (mod 2t). Then the search for orthogonal mappings is materially simplified by using a configuration based on the balanced incomplete block design with parameters v = b = 4t – 1, r = k = 2t – 1, λ = t – 1. Using this method, two sets of five mutually orthogonal Latin squares of order 12 were obtained.

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

This paper deals with the problem of finding sets of mutually orthogonal Latin squares of order 4t (where 4t – 1 is a prime power) based on orthogonal mappings of a group. For the group G we take the module G(2, 2t) whose elements are vectors (a 1, a 2,) where a 1 is a residue class (mod 2) and a 2 is a residue class (mod 2t), the addition being defined by (a 1, a 2) + (b 1, b 2) = (c 1, c 2) where a 1 + b 1 = C 1 (mod 2) and a 2 + b 2 = c 2 (mod 2t). Then the search for orthogonal mappings is materially simplified by using a configuration based on the balanced incomplete block design with parameters v = b = 4t – 1, r = k = 2t – 1, λ = t – 1. Using this method, two sets of five mutually orthogonal Latin squares of order 12 were obtained.

On the construction of sets of mutually orthogonal Latin squares and the falsity of a conjecture of Euler

then it is known, MacNeish [16] and Mann [17] that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. It seemed plausible that n(v) is also the maximum possible number of m.o.l.s. of order v. This would have implied the correctness of Euler's [13] conjecture about the nonexistence of two orthogonal Latin squares of order v when v =2 (mod 4), since n(v) -1 in this case. If we denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v, then Parker [18 ] showed that in certain cases N(v) >n(v) by proving that if there exists a balanced incomplete block (BIB) design with v treatments, X = 1, and block size k which is a prime power then N(v) > k -2. This result cast a doubt on the validity of Euler's conjecture. In this paper we generalize Parker's method of constructing m.o.l.s., by showing that a very general class of designs, which we have called pairwise balanced designs of index unity, can be used for the construction of sets of m.o.l.s. Various applications of this method have been made. In particular we show that Euler's conjecture is false for an infinity of values of v, including all values of the form 36m+22. We also provide a table for all values of v n(v). The smallest number for which we have been able to demonstrate the falsity of Euler's conjecture is 22. Two orthogonal squares of order 22 constructed by our method are given in the Appendix. Several attempts, necessarily erroneous, have been made in the past to prove Euler's conjecture, e.g., Peterson [19], Wernicke [20] and MacNeish [16]. Levi [14, p. 14] points

Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture

Ifis the prime power decomposition of an integer v, and we define the arithmetic function n(v) bythen it is known, MacNeish (10) and Mann (11), that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. We shall denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v. Then the Mann-MacNeish theorem can be stated asMacNeish conjectured that the actual value of N(v) is n(v).

On the Maximal Number of Pairwise Orthogonal Latin Squares of a Given Order

In the preceding paper Bose, Shrikhande, and Parker give their important discovery of the disproof of Euler's conjecture on Latin squares. In this paper we show that their results can be strengthened to imply that N(n), the maximal number of pairwise orthogonal Latin squares of order n, tends to infinity with n. In fact there exists a positive constant c, such that N(n) &gt; nc for all sufficiently large n.Our proof involves no new combinatorial insights, but is based entirely on a number-theoretical investigation of the following inequality due to Bose and Shrikhande.

Construction of some sets of mutually orthogonal latin squares

H. F. MacNeish [1 ] demonstrated constructively the existence of a set of t mutually orthogonal latin squares of each order n, where t is one less than the smallest factor of the prime-power decomposition of n. The construction was generalized somewhat and put on an algebraic foundation by H. B. Mann [2; 3, p. 105]. MacNeish [1] conjectured that t is the maximum number for each n. Had this conjecture been established, answers to two major questions would have been corollaries. These are: (1) the famous conjecture of Euler, dating from 1782, that there exists no pair of orthogonal latin squares of order= 2 (mod 4); (2) the conjecture that all finite projective planes are of prime-power orders-for an affine plane of order n is equivalent to a set of n -1 mutually orthogonal latin squares of order n. The purpose of this paper is to develop a construction yielding some new sets of mutually orthogonal latin squares. The general result is Theorem 1. For a few orders (possibly infinitely many distributed sparsely among the positive integers), Theorem 2 establishes the existence of sets of more than t mutually orthogonal latin squares; thus MacNeish's conjecture is disproved. Theorem 1 likely yields more than t for orders other than those covered by Theorem 2, but the author has found no example. The following lemma is familiar to some, but is apparently not in the literature.

On the Wiring of Two-Dimensional Multiple-Coincidence Magnetic Memories

The application of Minnick and Ashenhurst's technique of p-fold multiple-coincidence magnetic storage in an n x n array of cores is shown to require finding (p-2) orthogonal Latin squares of order n. The value of this technique lies in the reduction it effects in the disturbance of unselected cores. A method is suggested for reducing this disturbance to a level lower than that obtained by Minnick and Ashenhurst, by the application of reverse currents to the unselected interrogating wires during interrogation. The disturbance of unselected cores can be reduced to zero if p=n+1. In this case, (n+1) cores can be added to the store at the expense of a single additional interrogating wire. The resulting array of cores and interrogating wires is closely related to the finite projective geometry of order n.