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Sub-latin squares and incomplete orthogonal arrays

1974
*
Journal of combinatorial theory. Series A
*

The

doi:10.1016/0097-3165(74)90069-7
fatcat:c5jgwiaqvjfmpkkbxbosyvl3bm
*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. ...##
###
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. ...

##
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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

1980
*
Journal of combinatorial theory. Series A
*

The singular direct product employed in the construction of certain designs

doi:10.1016/0097-3165(80)90025-4
fatcat:74u34u5nmngnbbrlpjhec4skti
*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*. ...##
###
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

2004
*
Discrete Mathematics
*

Two

doi:10.1016/s0012-365x(03)00288-7
fatcat:pc64kx7uqjgnrhd3xsobel66he
*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*. ...##
###
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

1994
*
Journal of combinatorial theory. Series A
*

element of V occurs in precisely one cell of each row

doi:10.1016/0097-3165(94)90024-8
fatcat:rx6aisoc75cfzl5sen5trjovji
*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. ...##
###
Constructing ordered orthogonal arrays via sudoku

2016
*
Journal of Algebra and its Applications
*

However, Joint analysis of these experiments when conducted using

doi:10.1142/s0219498816501395
fatcat:zxan7nil35gxzhhmmwxrsmhyla
*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] ...##
###
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

2012
*
Bonfring International Journal of Data Mining
*

In this paper we have extended the Sudoku designs to

doi:10.9756/bijdm.1355
fatcat:tyctqdckhbaj3kl77tgt63l45u
*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. ...##
###
Existence of strong symmetric self-orthogonal diagonal Latin squares

2011
*
Discrete Mathematics
*

A diagonal

doi:10.1016/j.disc.2011.02.003
fatcat:5a5vyn4ewrepjat6xmklopbbjm
*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. ...##
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Orthogonal Latin squares: an application of experiment design to compiler testing

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.) ...

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