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Sets of mutually orthogonal Latin squares with "like subsquares"

1992
*
Journal of combinatorial theory. Series A
*

During this search, live additional

doi:10.1016/0097-3165(92)90052-v
fatcat:av5xdenu7re23mrzjrb7czuihy
*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. ...##
###
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]

2018
*
arXiv
*
pre-print

We have performed a complete enumeration

arXiv:1810.12639v1
fatcat:x7dfexkfjrgl5pimfyw7s56mbi
*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. ...##
###
Maximal sets of mutually orthogonal Latin squares

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*...

##
###
Recent results on Choi's orthogonal Latin squares

2022
*
Journal of Algebra Combinatorics Discrete Structures and Applications
*

There have been a few studies on Choi's

doi:10.13069/jacodesmath.1056511
fatcat:5dmt74l2tvez5l25ofg35znflm
*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). ...##
###
Recent results on Choi's orthogonal Latin squares
[article]

2021
*
arXiv
*
pre-print

There have been a few studies on Choi's

arXiv:1812.02202v3
fatcat:syjozmiadbahhlcljt6mtovkje
*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. ...##
###
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

2018
*
Physical Review A
*

We show that

doi:10.1103/physreva.97.062326
fatcat:mfggh7kxv5hsbf2uhykndljg6u
*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- ...##
###
A new representation of mutually orthogonal frequency squares
[article]

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] . ...

##
###
Enumeration of Sets of Mutually Orthogonal Latin Rectangles
[article]

2020
*
arXiv
*
pre-print

We study

arXiv:1910.02950v2
fatcat:bip5n53zcrhf7g3fc2x3h7st2y
*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. ...##
###
Quantum combinatorial designs and k-uniform states

2021
*
Journal of Physics A: Mathematical and Theoretical
*

In this article, we put forward

doi:10.1088/1751-8121/ac3705
fatcat:ifc7ldmq2rbwhkijdvegz6kg2u
*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. ...##
###
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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