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Sub-latin squares and incomplete orthogonal arrays

J.D Horton
1974 Journal of combinatorial theory. Series A  
The orthogonal array corresponding to a set of pairwise orthogonal Latin squares, minus the subarray corresponding to orthogonal sub-squares is called an incomplete orthogonal array; this concept is generalized  ...  sub-squares.  ...  Each square of a set of s pairwise orthogonal latin squares of order n (sPOLS(n)) may have a Latin sub-square of size k and each sub-square may appear in the same set of rows and columns.  ... 
doi:10.1016/0097-3165(74)90069-7 fatcat:c5jgwiaqvjfmpkkbxbosyvl3bm

Page 5590 of Mathematical Reviews Vol. , Issue 90J [page]

1990 Mathematical Reviews  
Latin squares with some additional properties including orthogonal subsquares, conjugate orthogonality, orthogonal diagonal Latin squares and orthogonal Latin squares with holes are discussed.  ...  Construction of a set of K pairwise orthogonal Latin squares from a given n x (n + 1) circular Florentine array is illustrated for n= 4.  ... 

Page 4428 of Mathematical Reviews Vol. , Issue 86j [page]

1986 Mathematical Reviews  
Wang [“On self-orthogonal Latin squares and partial transverals of Latin squares”, Ph.D.  ...  A pair of orthogonal diagonal Latin squares is a pair of orthogonal Latin squares such that each has a transversal on both the main diagonal and the back diagonal.  ... 

A generalization of the singular direct product with applications to skew Room squares

R.C Mullin
1980 Journal of combinatorial theory. Series A  
The singular direct product employed in the construction of certain designs and quasigroups, is generalized.  ...  When applied to skew Room squares, it extends earlier results to show that the spectrum is complete for values above 237. (skew) Room square of side v2 containing a (skew) subsquare of side vX (if 306  ...  To do so we require certain incomplete arrays. Fortunately there is a direct singular product for incomplete arrays and orthogonal latin squares.  ... 
doi:10.1016/0097-3165(80)90025-4 fatcat:74u34u5nmngnbbrlpjhec4skti

Page 21 of Mathematical Reviews Vol. 50, Issue 1 [page]

1975 Mathematical Reviews  
D. 143 Sub-Latin squares and incomplete orthogonal arrays. J. Combinatorial Theory Ser. A 16 (1974), 23-33.  ...  The orthogonal array corre- sponding to a set of pairwise orthogonal Latin squares, minus the sub-array corresponding to orthogonal sub-squares, is called 50 #4143-146 elongate orthogonal array; this concept  ... 

On the spectrum of r-self-orthogonal Latin squares

Yunqing Xu, Yanxun Chang
2004 Discrete Mathematics  
Two Latin squares of order n are r-orthogonal if their superposition produces exactly r distinct ordered pairs.  ...  |{(l ij ; m ij ): 0 6 i; j 6 n − 1}| = r: Belyavskaya (see [2] [3] [4] ) ÿrst systematically treated the following question: For which integers n and r does a pair of r-orthogonal Latin squares of order  ...  If H = {S 1 }, we simply write ISOLS(n; s 1 ) for the incomplete self-orthogonal Latin square.  ... 
doi:10.1016/s0012-365x(03)00288-7 fatcat:pc64kx7uqjgnrhd3xsobel66he

Page 4030 of Mathematical Reviews Vol. , Issue 87h [page]

1987 Mathematical Reviews  
Denote by (7,7,4)-COILS(v) an idempotent Latin square of or- der v which is orthogonal to its (7,7, k)-conjugate and by (2,7, k)- ICOILS(v,n) an incomplete (i,7,k)-COILS(v) missing a sub- COILS(n), where  ...  [Zhu, Lie] (PRC-SOO) On the existence of incomplete conjugate orthogonal idempotent Latin squares. Tenth British combinatorial conference (Glasgow, 1985). Ars Combin. 20 (1985), A, 193-210.  ... 

Howell designs with sub-designs

J.H Dinitz, E.R Lamken
1994 Journal of combinatorial theory. Series A  
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.  ...  A Howell design of side s and order 2n, or more briefly an H(s, 2n), is an s x s array in which each cell is either empty or contains an unordered pair of elements from some (2n)-set V such that (1) every  ...  The existence of Room squares with Room square subdesigns and a pair of mutually orthogonal Latin squares with Latin square sub-designs has been investigated.  ... 
doi:10.1016/0097-3165(94)90024-8 fatcat:rx6aisoc75cfzl5sen5trjovji

Constructing ordered orthogonal arrays via sudoku

John Lorch
2016 Journal of Algebra and its Applications  
However, Joint analysis of these experiments when conducted using orthogonal (Graeco) Sudoku square design is still missing in the literature.  ...  treatments and periods (seasons).  ...  Sudoku square design consists of treatments (Latin letters) which are arranged in a square array in such a way that each row, column and sub-square contains each of the treatments exactly once see [8]  ... 
doi:10.1142/s0219498816501395 fatcat:zxan7nil35gxzhhmmwxrsmhyla

Page 2446 of Mathematical Reviews Vol. , Issue 94e [page]

1994 Mathematical Reviews  
Some new results on orthogonal incomplete arrays and balanced incom- plete arrays are given.  ...  (Latin square with quadrangle criterion) to have an orthogonal mate.  ... 

Page 43 of Mathematical Reviews Vol. , Issue 87a [page]

1987 Mathematical Reviews  
The author considers the question of when a Latin square L of order n can be embedded in a complete set of mutually orthogonal Latin squares.  ...  A reduced Latin square has the “rst row and column in the standard order 1,2,---,n.  ... 

Page 5443 of Mathematical Reviews Vol. , Issue 86m [page]

1986 Mathematical Reviews  
Maurin, Francis On incomplete orthogonal arrays. J. Combin. Theory Ser. A 40 (1985), no. 1, 183-185.  ...  C. 86m:05023 Mutually orthogonal partitions of the 6 x 6 Latin squares. Utilitas Math. 27 (1985), 265-274.  ... 

Construction of Graeco Sudoku Square Designs of Odd Orders

Subramani J
2012 Bonfring International Journal of Data Mining  
In this paper we have extended the Sudoku designs to Orthogonal (Graeco) Sudoku square designs in line with that of the Orthogonal (Graeco) latin square designs.  ...  The Sudoku designs are similar to that of latin square designs but accommodate some additional factors.  ...  That is, the problem of constructing a Graeco Latin square and a pair of orthogonal Latin squares are one and the same.  ... 
doi:10.9756/bijdm.1355 fatcat:tyctqdckhbaj3kl77tgt63l45u

Existence of strong symmetric self-orthogonal diagonal Latin squares

H. Cao, W. Li
2011 Discrete Mathematics  
A diagonal Latin square is a Latin square whose main diagonal and back diagonal are both transversals. A Latin square is self-orthogonal if it is orthogonal to its transpose.  ...  A diagonal Latin square L of order n is strongly symmetric, denoted by SSSODLS(n), if In this note, we shall prove that an SSSODLS(n) exists if and only if n ≡ 0, 1, 3 (mod 4) and n ̸ = 3.  ...  An incomplete SOLS is a self-orthogonal Latin square of order n missing a sub-SOLS of order k, denoted by ISOLS(n, k). For the existence of an ISOLS(n, k), see [1, 9, 7, 8, 10] . Theorem 2.1.  ... 
doi:10.1016/j.disc.2011.02.003 fatcat:5a5vyn4ewrepjat6xmklopbbjm

Orthogonal Latin squares: an application of experiment design to compiler testing

Robert Mandl
1985 Communications of the ACM  
Latin squares and balanced incomplete block designs.  ...  (The existence of a pair of orthogonal Latin squares of some order does not imply that for any Latin square of that order one can find a Latin square orthogonal to it.)  ... 
doi:10.1145/4372.4375 fatcat:j26cqi5fjranfo7ldwdpuo4x24
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