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A few more r-orthogonal latin squares

L. Zhu, H. Zhang
2001 Discrete Mathematics  
Two Latin squares are r-orthogonal if their superposition produces r distinct pairs.  ...  It was Belyavskaya who ÿrst systematically treated the following question: For which integers n and r does a pair of r-orthogonal Latin squares of order n exist?  ...  (see [1] [2] [3] ), who ÿrst systematically treated the following question: For which integers n and r does a pair of r-orthogonal Latin squares of order n exist?  ... 
doi:10.1016/s0012-365x(00)00424-6 fatcat:fthdr3qqxfgavhdzfnv65z6cji

Distance-Constrained Orthogonal Latin Squares for Brain-Computer Interface

Gang Luo, Wanli Min
2010 Journal of medical systems  
However, it is unknown whether such distance-constrained, orthogonal Latin square pairs actually exist.  ...  These orthogonal Latin square pairs satisfy certain distance constraint so that neighboring characters are not intensified simultaneously.  ...  the method described in Section 2. 26 characters, 10 digits, and a few symbols require more than 36 matrix cells but no more than 81 matrix cells.  ... 
doi:10.1007/s10916-010-9455-6 pmid:20703739 fatcat:5z5vftizrrgjlamto3lc4iouny

Page 946 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 10, Issue 6 [page]

1959 American Mathematical Society. Proceedings of the American Mathematical Society  
For a few orders (possibly infinitely many distributed sparsely among the positive integers), Theorem 2 establishes the existence of sets of more than ¢ mutually orthogonal latin squares; thus MacNeish  ...  A set of k—2 mutually orthogonal latin squares of order n is equivalent to a set of n* ordered k-tuples, (a, @i2, - - - ,@ix),t=1, - + - ,n?  ... 

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 also give a new way to construct magic squares from two orthogonal non-double-diagonal Latin squares, which explains why Choi's Latin squares produce a magic square of order 9.  ...  If we take three or more elements of M from the set {r 0 (A), r 1 (A), r 2 (A), r 3 (A)}, then there should appear a pair of non-orthogonal Latin squares.  ... 
doi:10.13069/jacodesmath.1056511 fatcat:5dmt74l2tvez5l25ofg35znflm

Upper Bounds on Sets of Orthogonal Colorings of Graphs [article]

Serge C. Ballif
2012 arXiv   pre-print
latin squares, or sudoku squares are a special cases of bounds on orthogonal colorings of graphs.  ...  We show that the usual bounds on the maximum size of a certain set of orthogonal latin structures such as latin squares, row latin squares, equi-n squares, single diagonal latin squares, double diagonal  ...  No set of mutually orthogonal row (or column) latin squares of order n consists 142 of more than n squares. No set of mutually orthogonal latin squares consists of more than n − 1 squares.  ... 
arXiv:1110.2237v3 fatcat:wz5a3m2tgrhxnjh7imofwe2lsy

Finding orthogonal latin squares using finite model searching tools

FeiFei Ma, Jian Zhang
2011 Science China Information Sciences  
This paper describes how to use a general-purpose model searching program to find orthogonal latin squares.  ...  An important class of problems in combinatorics is to find orthogonal latin squares with certain properties. Computer search is a promising approach for solving such problems.  ...  More formally, two latin squares L 1 and L 2 are said to be orthogonal if for any x, y ∈ R, and z, w ∈ C, (L 1 (x, z) = L 1 (y, w) ∧ L 2 (x, z) = L 2 (y, w)) → (x = y ∧ z = w).  ... 
doi:10.1007/s11432-011-4343-3 fatcat:6c7rrkdod5anfi7siwwkg5mnca

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 also give a new way to construct magic squares from two orthogonal non-double-diagonal Latin squares, which explains why Choi's Latin squares produce a magic square of order 9.  ...  To prove a more general statement, we include the proof of the above theorem. Proof. Suppose that A and B, the pair of orthogonal diagonal Latin squares of order n are given.  ... 
arXiv:1812.02202v3 fatcat:syjozmiadbahhlcljt6mtovkje

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

2003 Mathematical Reviews  
|Zhang, Hantao] (1-IA-C; Iowa City, IA) A few more r-orthogonal Latin squares. (English summary) Designs, codes and finite geometries (Shanghai, 1999). Discrete Math. 238 (2001), no. 1-3, 183-191.  ...  Summary: “Two Latin squares are r-orthogonal if their super- position produces r distinct pairs.  ... 

Orthogonal Latin Square Configuration for LSI Memory Yield and Reliability Enhancement

M.Y. Hsiao, D.C. Bossen
1975 IEEE transactions on computers  
k different words using orthogonal Latin squares of order 2r Fig. 4 illustrates the idea of the proof by assuming that has high probability of success in a few trys. [4] R.  ...  Bossen, and R. T. Chien, "Orthogonal Latin square codes," IBM J. Res. Develop., vol. 14, no. orthogonal Latin square each word now has a single error.  ... 
doi:10.1109/t-c.1975.224254 fatcat:5d2sii7d2zfzldldogdtpo7xpu

A connection between random variables and latin k-cubes

Ruben Michel, Gadi Taubenfeld, Andrew Berman
1995 Discrete Mathematics  
The subject of latin squares is about 200 years old, and it abounds with many solved and unsolved problems. In this paper we establish an interconnection between latin k-cubes and random variables.  ...  When combined with the rich theory of latin k-cubes, this connection yields new results about independent random variables, which generalize and extend other recent results.  ...  It is known that for prime p there exist p -1 orthogonal latin squares and no more.  ... 
doi:10.1016/0012-365x(94)00073-7 fatcat:77lcajka4vhxjb3joen7ykmqnu

Page 8403 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
[Zhu, Lie'] (PRC-SOO; Suzhou) A few more incomplete self-orthogonal Latin squares and related designs. (English summary) Australas. J. Combin. 21 (2000), 85-94.  ...  Summary: “An incomplete self-orthogonal Latin square of order v with an empty subarray of order n, an ISOLS(v, n), can exist only if v > 3n+1.  ... 

Construction of Some Sets of Mutually Orthogonal Latin Squares

E. T. Parker
1959 Proceedings of the American Mathematical Society  
prime-power orders-for an affine plane of order re is equivalent to a set of re -1 mutually orthogonal latin squares of order re.  ...  demonstrated constructively the existence of a set of t mutually orthogonal latin squares of each order re, where / is one less than the smallest factor of the prime-power decomposition of re.  ...  Robert Silverman of the Ohio State University for pointing out that in Theorem 1 the system of permutations need not be a group.  ... 
doi:10.2307/2033627 fatcat:zqtj7qfxtfdixgewsn6zf2o7rq

Construction of some sets of mutually orthogonal latin squares

E. T. Parker
1959 Proceedings of the American Mathematical Society  
prime-power orders-for an affine plane of order re is equivalent to a set of re -1 mutually orthogonal latin squares of order re.  ...  demonstrated constructively the existence of a set of t mutually orthogonal latin squares of each order re, where / is one less than the smallest factor of the prime-power decomposition of re.  ...  Robert Silverman of the Ohio State University for pointing out that in Theorem 1 the system of permutations need not be a group.  ... 
doi:10.1090/s0002-9939-1959-0109789-9 fatcat:dolddxmplrc2dbmu4it4rro424

Page 4144 of Mathematical Reviews Vol. , Issue 92h [page]

1992 Mathematical Reviews  
Heinrich); (5) Recursive con- structions of mutually orthogonal Latin squares (A. E. Brouwer); (6) r-orthogonal Latin squares (G. B. Belyavskaya); (7) Latin squares and universal algebra (T.  ...  In this paper the author obtains a necessary con- dition and a sufficient condition for constructing double magic squares by using orthogonal Latin squares, and constructs the smallest double magic squares  ... 

Some Relations on Paratopisms and An Intuitive Interpretation on the Adjugates of a Latin Square [article]

Wenwei Li
2018 arXiv   pre-print
These methods could distinctly simplify the computation on a computer for the issues related to main classes of Latin squares.  ...  With this trick, we can generate the adjugates of arbitrary Latin square directly from the original one without generating the orthogonal array.  ...  In 2012, when the author read an old paper on Latin square and LYaPAS related to this article written in Russian by Prof.  ... 
arXiv:1803.02196v2 fatcat:dzavqzphifelpn54l5a7rms52a
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