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Sets of mutually orthogonal Latin squares with "like subsquares"

Charles E Roberts
1992 Journal of combinatorial theory. Series A  
During this search, live additional sets of four mutually orthogonal Latin squares were discovered. These sets are maximal in the usual sense.  ...  One such set consists of S,, and the four Latin squares displayed in Fig. 1 S, and D, are two mutually orthogonal Latin squares of order 3.  ... 
doi:10.1016/0097-3165(92)90052-v fatcat:av5xdenu7re23mrzjrb7czuihy

Page 622 of Mathematical Reviews Vol. 21, Issue 6 [page]

1960 Mathematical Reviews  
/4 ordered quadruples equivaient to a pair of orthogonal Latin squares of order (q—1)/2, with the X; in all four positions.  ...  Whereas Parker proves that the existence of a balanced incomplete block design with A=1 and k a prime power implies the existence of a set of k — 2 pairwise orthogonal Latin squares of order v, the authors  ... 

Triples of Orthogonal Latin and Youden Rectangles For Small Orders [article]

Gerold Jäger, Klas Markström, Lars-Daniel Öhman, Denys Shcherbak
2018 arXiv   pre-print
We have performed a complete enumeration of non-isotopic triples of mutually orthogonal k× n Latin rectangles for k≤ n ≤ 7.  ...  We have also studied orthogonal triples of k × 8 rectangles which are formed by extending mutually orthogonal triples with non-trivial autotopisms one row at a time, and requiring that the autotopism group  ...  This work was supported by the Swedish strategic research programme eSSENCE. This work was supported by The Swedish Research Council grant 2014-4897. Algorithm 1: Extension of a triple of MOLR.  ... 
arXiv:1810.12639v1 fatcat:x7dfexkfjrgl5pimfyw7s56mbi

Maximal sets of mutually orthogonal Latin squares

David A. Drake, G.H.J. van Rees, W.D. Wallis
1999 Discrete Mathematics  
Maximal sets of s mutually orthogonal Latin squares of order v are constructed for infinitely many new pairs (s,v).  ...  Introduction A set S of mutually orthogonal Latin squares (MOLS) is maximal if no Latin square is orthogonal to each member of S.  ...  Using known results on the existence of sets of MOLS (see, e.g., [1] ), we obtain the following conclusions: Pairs of orthogonal Latin squares without common orthogonal mates There is a Latin square  ... 
doi:10.1016/s0012-365x(98)00119-8 fatcat:tq3bb6pl4bhxblfi3j42t44yje

Recent results on Choi's orthogonal Latin squares

Jon-lark KİM, Dong Eun OHK, Doo Young PARK, Jae Woo PARK
2022 Journal of Algebra Combinatorics Discrete Structures and Applications  
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 × 3 orthogonal Latin squares.  ...  We find a geometric description of Choi's orthogonal Latin squares of order 9 using the dihedral group D8.  ...  Let M be the set of mutually orthogonal Latin squares, which has the maximum number of mutually orthogonal Latin squares in the set D 8 (A).  ... 
doi:10.13069/jacodesmath.1056511 fatcat:5dmt74l2tvez5l25ofg35znflm

Recent results on Choi's orthogonal Latin squares [article]

Jon-Lark Kim, Dong Eun Ohk, Doo Young Park, Jae Woo Park
2021 arXiv   pre-print
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 × 3 orthogonal Latin squares.  ...  We find a geometric description of Choi's orthogonal Latin squares of order 9 using the dihedral group D_8.  ...  Then the maximum number of mutually orthogonal Latin squares of order n in the set D 8 (A) is 4.  ... 
arXiv:1812.02202v3 fatcat:syjozmiadbahhlcljt6mtovkje

Page 51 of Mathematical Reviews Vol. , Issue 2003A [page]

2003 Mathematical Reviews  
In particular, we provide a new proof for the bound on the maximal size of a set of MOFS and obtain a geometric characterisation of the case of equality: neces- sary and sufficient conditions for the existence  ...  It was Belyavskaya who first systematically treated the following question: For which integers n and r does a pair of r-orthogonal Latin squares of order n ex- ist?  ... 

Page 4 of Mathematical Reviews Vol. 32, Issue 4 [page]

1966 Mathematical Reviews  
Let N(s) denote the maximal number of mutually orthog- onal latin squares of order s.  ...  This paper gives a survey of the currently known results concerning N(s), and a con- struction of a complete set of s—1 mutually orthogonal latin squares of order s from a given set of s— 2 mutually orthogonal  ... 

Entanglement and quantum combinatorial designs

Dardo Goyeneche, Zahra Raissi, Sara Di Martino, Karol Życzkowski
2018 Physical Review A  
We show that mutually orthogonal quantum Latin arrangements can be entangled in the same way than quantum states are entangled.  ...  We introduce several classes of quantum combinatorial designs, namely quantum Latin squares, cubes, hypercubes and a notion of orthogonality between them.  ...  G. and K.Ż. are supported by the Narodowe Centrum Nauki under the project num-  ... 
doi:10.1103/physreva.97.062326 fatcat:mfggh7kxv5hsbf2uhykndljg6u

A new representation of mutually orthogonal frequency squares [article]

Jonathan Jedwab, Tabriz Popatia
2020 arXiv   pre-print
Mutually orthogonal frequency squares (MOFS) of type F(mλ;λ) generalize the structure of mutually orthogonal Latin squares: rather than each of m symbols appearing exactly once in each row and in each  ...  Wanless, Mutually orthogonal binary frequency squares, Electron. J. Combin., 27(#P3.7), 2020, 26 pages] of the case when m=2 and λ is odd.  ...  sharing extensive data on MOFS as well as a preliminary version of [3] .  ... 
arXiv:2003.03920v2 fatcat:q7z73fpavrhcjdmfjcorm5vum4

Enumeration of Sets of Mutually Orthogonal Latin Rectangles [article]

Gerold Jäger, Klas Markström, Denys Shcherbak, Lars-Daniel Öhman
2020 arXiv   pre-print
We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural variation of the concept of self-orthogonal Latin squares which is applicable on larger sets of mutually orthogonal Latin squares  ...  We call such a set of MOLR homogeneous. In the course of doing this, we perform a complete enumeration of non-isotopic sets of t mutually orthogonal k× n Latin rectangles for k≤ n ≤ 7, for all t < n.  ...  We would also like to thank the anonymous reviewer for pointing out the connection to orthogonal arrays mentioned in the last section.  ... 
arXiv:1910.02950v2 fatcat:bip5n53zcrhf7g3fc2x3h7st2y

Quantum combinatorial designs and k-uniform states

Yajuan Zang, Paolo Facchi, Zihong Tian
2021 Journal of Physics A: Mathematical and Theoretical  
In this article, we put forward the notions of incomplete quantum Latin squares and orthogonality on them and present construction methods for mutually orthogonal quantum Latin squares and mutually orthogonal  ...  Furthermore, we introduce the notions of generalized mutually orthogonal quantum Latin squares and generalized mutually orthogonal quantum Latin cubes, which are equivalent to quantum orthogonal arrays  ...  Zang acknowledges the hospitality of the PhD school in Physics at the Physics Department of the University of Bari.  ... 
doi:10.1088/1751-8121/ac3705 fatcat:ifc7ldmq2rbwhkijdvegz6kg2u

Page 7818 of Mathematical Reviews Vol. , Issue 2002K [page]

2002 Mathematical Reviews  
We also give an example of two mutually complementary critical sets, which partition a Latin square of order 8 into two disjoint critical sets. 2002k:05042 OSBIS5 62K15 Collombier, Dominique (F-STRAS;  ...  We give a theorem to show that for a given order n, there exists a back- circulant Latin square of order n which may be partitioned into four disjoint critical sets, and we give examples of all possible  ... 

Page 7322 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
Summary: “Mutually orthogonal sets of hypercubes are higher- 05 COMBINATORICS 7322 dimensional generalizations of mutually orthogonal sets of Latin squares.  ...  For Latin squares, it is well known that the Cayley table of a group of order n is a Latin square, which has no orthogonal mate if nm is congruent to 2 modulo 4.  ... 

Page 6449 of Mathematical Reviews Vol. , Issue 96k [page]

1996 Mathematical Reviews  
We first survey construc- tions for maximal sets of mutually orthogonal Latin squares of orders m < 32 which use orthomorphisms of abelian groups and then show how several recent constructions implicitly  ...  Summary: “Orthomorphisms of finite groups have several combi- natorial applications—for instance in the construction of designs and in the construction of mutually orthogonal sets of Latin squares.  ... 
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