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Journal articles on the topic 'Latin squares'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

M., I. Garcia-Planas, and Roca-Borrego D. "Cyclic-union Operation to Obtain Latin Squares." British Journal of Mathematics & Computer Science 22, no. 5 (2017): 1–8. https://doi.org/10.9734/BJMCS/2017/33945.

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With the power that has taken the information technologies, one has developed the study and research about cryptography, and cryptanalysis, in which Latin squares are ideal candidates for being used in cryptographic systems because the Cayley tables of the nite groups are Latin squares. This fact has awakened a new interest in the study of Latin squares by applying them to the study of code theory and error correcting codes. They also play a significant role in the statistical theory of experimental design. In this work, we develop an algorithm for the generation of Latin squares based on the

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2

Bagchi, Bhaskar. "Latin squares." Resonance 17, no. 9 (2012): 895–902. http://dx.doi.org/10.1007/s12045-012-0098-4.

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3

Fearn, Tom. "Latin Squares." NIR news 17, no. 1 (2006): 15. http://dx.doi.org/10.1255/nirn.875.

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4

Yu, Glebsky L., and Carlos J. Rubio. "Latin Squares, Partial Latin Squares and Their Generalized Quotients." Graphs and Combinatorics 21, no. 3 (2005): 365–75. http://dx.doi.org/10.1007/s00373-005-0614-3.

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5

Emanouilidis, Emanuel. "Latin and cross Latin squares." International Journal of Mathematical Education in Science and Technology 39, no. 5 (2008): 697–700. http://dx.doi.org/10.1080/00207390801935921.

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6

Bailey, R. A. "Semi-Latin squares." Journal of Statistical Planning and Inference 18, no. 3 (1988): 299–312. http://dx.doi.org/10.1016/0378-3758(88)90107-3.

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7

Cavenagh, Nicholas, Carlo Hämäläinen, James G. Lefevre, and Douglas S. Stones. "Multi-latin squares." Discrete Mathematics 311, no. 13 (2011): 1164–71. http://dx.doi.org/10.1016/j.disc.2010.06.026.

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8

Danziger, Peter, Ian M. Wanless, and Bridget S. Webb. "Monogamous latin squares." Journal of Combinatorial Theory, Series A 118, no. 3 (2011): 796–807. http://dx.doi.org/10.1016/j.jcta.2010.11.011.

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9

Heinrich, Katherine, Kichul Kim, and V. K. Prasanna Kumar. "Perfect latin squares." Discrete Applied Mathematics 37-38 (July 1992): 281–86. http://dx.doi.org/10.1016/0166-218x(92)90139-2.

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10

Jenkins, Peter. "Embedding a latin square in a pair of orthogonal latin squares." Journal of Combinatorial Designs 14, no. 4 (2006): 270–76. http://dx.doi.org/10.1002/jcd.20087.

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11

Katrnoška, František. "On algebras of generalized Latin squares." Mathematica Bohemica 136, no. 1 (2011): 91–103. http://dx.doi.org/10.21136/mb.2011.141453.

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12

Falcón, Raúl M., Víctor Álvarez, José Andrés Armario, María Dolores Frau, Félix Gudiel, and María Belén Güemes. "A computational approach to analyze the Hadamard quasigroup product." Electronic Research Archive 31, no. 6 (2023): 3245–63. http://dx.doi.org/10.3934/era.2023164.

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<abstract><p>Based on the binary product described by any Latin square, the Hadamard quasigroup product is introduced in this paper as a natural generalization of the classical Hadamard product of matrices. The successive iteration of this new product is endowed with a cyclic behaviour that enables one to define a pair of new isomorphism invariants of Latin squares. Of particular interest is the set of Latin squares for which this iteration preserves the Latin square property, which requires the existence of successive localized Latin transversals within the Latin square under cons

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13

EDMONDSON, R. N. "Trojan square and incomplete Trojan square designs for crop research." Journal of Agricultural Science 131, no. 2 (1998): 135–42. http://dx.doi.org/10.1017/s002185969800567x.

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Latin square and near-Latin square designs are valuable row-and-column designs for crop research but the practical size range of such designs is severely limited. Semi-Latin square designs extend this range but not all semi-Latin squares are suitable for experimental designs. Trojan square designs are a special class of optimal semi-Latin squares that generalizes the class of Latin square designs. The construction of Trojan squares both for unstructured and for factorial treatment sets is discussed and the utility of Trojan square designs for practical crop research is demonstrated. The corpus

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14

Kuhl, Jaromy Scott, and Tristan Denley. "On avoiding odd partial Latin squares and r-multi Latin squares." Discrete Mathematics 306, no. 22 (2006): 2968–75. http://dx.doi.org/10.1016/j.disc.2006.05.028.

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15

Mahfooz, S. Z., and Y. Khan. "Characteristics of Dispersed Latin Squares." Nucleus 56, no. 4 (2020): 131–36. https://doi.org/10.71330/thenucleus.2019.585.

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Experimental designs widely use Latin squares for controlled testing to understand causal processes.Among many existing Latin squares for a given order, a design of experiment may need a specific Latinsquare that offers its treatments (symbols) scattered at the desired locations. There is a need to studysystematic dispersion of symbols in Latin squares to assist effective research designs. This paperdescribes the attributes and applications of one of such designs known as Latin square of diamonddispersed pattern

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16

Han, Michael, Ella Kim, Evin Liang, et al. "Fun with Latin Squares." Recreational Mathematics Magazine 10, no. 17 (2023): 51–74. http://dx.doi.org/10.2478/rmm-2023-0003.

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Abstract Do you want to know what an anti-chiece Latin square is? Or what a non-consecutive toroidal modular Latin square is? We invented a ton of new types of Latin squares, some inspired by existing Sudoku variations. We can’t wait to introduce them to you and answer important questions, such as: do they even exist? If so, under what conditions? What are some of their interesting properties? And how do we generate them?

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17

Abbas, Fazal, Mubasher Umer, Umar Hayat, and Ikram Ullah. "Trivial and Nontrivial Eigenvectors for Latin Squares in Max-Plus Algebra." Symmetry 14, no. 6 (2022): 1101. http://dx.doi.org/10.3390/sym14061101.

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A square array whose all rows and columns are different permutations of the same length over the same symbol set is known as a Latin square. A Latin square may or may not be symmetric. For classification and enumeration purposes, symmetric, non-symmetric, conjugate symmetric, and totally symmetric Latin squares play vital roles. This article discusses the Eigenproblem of non-symmetric Latin squares in well known max-plus algebra. By defining a certain vector corresponding to each cycle of a permutation of the Latin square, we characterize and find the Eigenvalue as well as the possible Eigenve

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18

FU, CHIN-MEI. "A NOTE ON THE CONSTRUCTION OF LARGE SET OF LATIN SQUARES WITH ONE ENTRY IN COMMON." Tamkang Journal of Mathematics 24, no. 2 (1993): 215–20. http://dx.doi.org/10.5556/j.tkjm.24.1993.4492.

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   A latin square of order $n$ is an $n \times n$ array such that each of the integers $1, 2, 3, \cdots, n$ occurs exactly once in each row and each column. A large set of latin squares of order $n$ having only one entry in common is a maximum set of latin squares of order $n$ such that each pair of them contains exactly one fixed entry in common. In this paper, we prove that a large set of latin squares of order $n$ having only one entry in common has $n - 1$ latin squares for each positive integer $n$, $n \ge 4$.   

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19

Kim, Beob G., and Hans H. Stein. "A spreadsheet program for making a balanced Latin Square design." Revista Colombiana de Ciencias Pecuarias 22, no. 4 (2009): 6. http://dx.doi.org/10.17533/udea.rccp.324493.

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Summary Latin square designs are often employed in animal experiments to minimize the number of animals required to detect statistical differences. Generally, potential carryover effects are not balanced out by randomization. Systemic methods are available for equalizing the residual effects. We have developed an Excel® spreadsheet-based program, the Balanced Latin Square Designer (BLSD), to facilitate the generation of Latin squares balanced for carryover effects. The program allows a user to input the number of treatments that is equal to the number of animals and periods in a square. A user

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20

Behforooz, Hossein. "91.49 Mirror magic squares from Latin Squares." Mathematical Gazette 91, no. 521 (2007): 316–21. http://dx.doi.org/10.1017/s0025557200181793.

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21

Faruqi, Shahab, S. A. Katre, and Manisha Garg. "Pseudo orthogonal Latin squares." Discrete Mathematics and Applications 31, no. 1 (2021): 5–17. http://dx.doi.org/10.1515/dma-2021-0002.

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Abstract Two Latin squares A, B of order n are called pseudo orthogonal if for any 1 ≤ i, j ≤ n there exists a k, 1 ≤ k ≤ n, such that A(i, k) = B(j, k). We prove that the existence of a family of m mutually pseudo orthogonal Latin squares of order n is equivalent to the existence of a family of m mutually orthogonal Latin squares of order n. We also obtain exact values of clique partition numbers of several classes of complete multipartite graphs and of the tensor product of complete graphs.

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22

Galatenko, A. V., V. A. Nosov, and A. E. Pankratiev. "Latin Squares over Quasigroups." Lobachevskii Journal of Mathematics 41, no. 2 (2020): 194–203. http://dx.doi.org/10.1134/s1995080220020079.

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23

Emanouilidis, Emanuel. "Latin and magic squares." International Journal of Mathematical Education in Science and Technology 36, no. 5 (2005): 546–49. http://dx.doi.org/10.1080/00207390412331336201.

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24

Pittenger, Arthur O. "Mappings of latin squares." Linear Algebra and its Applications 261, no. 1-3 (1997): 251–68. http://dx.doi.org/10.1016/s0024-3795(96)00408-9.

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25

Hilton, A. J. W., M. Mays, C. A. Rodger, and C. St J. A. Nash-Williams. "Hamiltonian double latin squares." Journal of Combinatorial Theory, Series B 87, no. 1 (2003): 81–129. http://dx.doi.org/10.1016/s0095-8956(02)00029-1.

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26

Easton, T., and R. Gary Parker. "On completing latin squares." Discrete Applied Mathematics 113, no. 2-3 (2001): 167–81. http://dx.doi.org/10.1016/s0166-218x(00)00282-1.

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27

Krafft, Olaf, Herbert Pahlings, and Martin Schaefer. "Diagonal-complete latin squares." European Journal of Combinatorics 24, no. 3 (2003): 229–37. http://dx.doi.org/10.1016/s0195-6698(03)00025-8.

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28

Häggkvist, R. "All-even latin squares." Discrete Mathematics 157, no. 1-3 (1996): 127–46. http://dx.doi.org/10.1016/0012-365x(95)00263-v.

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29

Shen, Xiaojun. "Generalized Latin squares II." Discrete Mathematics 143, no. 1-3 (1995): 221–42. http://dx.doi.org/10.1016/0012-365x(95)98135-s.

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30

Chetwynd, A. G., and A. J. W. Hilton. "Outline symmetric latin squares." Discrete Mathematics 97, no. 1-3 (1991): 101–17. http://dx.doi.org/10.1016/0012-365x(91)90426-3.

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31

Hilton, Anthony J. W., and Jerzy Wojciechowski. "Amalgamating infinite latin squares." Discrete Mathematics 292, no. 1-3 (2005): 67–81. http://dx.doi.org/10.1016/j.disc.2003.09.016.

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32

Woodcock, C. F. "On orthogonal latin squares." Journal of Combinatorial Theory, Series A 43, no. 1 (1986): 146–48. http://dx.doi.org/10.1016/0097-3165(86)90034-8.

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33

Häggkvist, Roland, and Jeannette C. M. Janssen. "All-even latin squares." Discrete Mathematics 157, no. 1-3 (1996): 199–206. http://dx.doi.org/10.1016/s0012-365x(96)83015-9.

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34

Lakic, Nada. "Classification of Latin squares." Journal of Agricultural Sciences, Belgrade 47, no. 1 (2002): 105–12. http://dx.doi.org/10.2298/jas0201105l.

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Efficacy and profitability of results and eventually the conclusions of an experiment were found to depend on the statistical model for organizing an experiment. No thoroughgoing studies have been reported to date in our statistical literature on Latin square designs, one of the three basic experimental designs. The objective of the study was to define the insufficiently known subsets of Latin square designs having special properties and classify them using a number of criteria.

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35

Shen, Xiaojun, Y. Z. Cai, C. L. Liu, and Clyde P. Kruskal. "Generalized latin squares I." Discrete Applied Mathematics 25, no. 1-2 (1989): 155–78. http://dx.doi.org/10.1016/0166-218x(89)90052-8.

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36

Wanless, Ian M. "Diagonally cyclic latin squares." European Journal of Combinatorics 25, no. 3 (2004): 393–413. http://dx.doi.org/10.1016/j.ejc.2003.09.014.

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37

Marini, A., and G. Pirillo. "Signs on Latin Squares." Advances in Applied Mathematics 15, no. 4 (1994): 490–505. http://dx.doi.org/10.1006/aama.1994.1021.

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38

Kwan, Matthew, Ashwin Sah, Mehtaab Sawhney, and Michael Simkin. "Substructures in Latin squares." Israel Journal of Mathematics 256, no. 2 (2023): 363–416. http://dx.doi.org/10.1007/s11856-023-2513-9.

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39

Kirichenko, V. V., M. A. Khibina, V. M. Zhuravlev, and O. V. Zelensky. "Quivers and Latin squares." São Paulo Journal of Mathematical Sciences 10, no. 2 (2015): 286–300. http://dx.doi.org/10.1007/s40863-015-0031-3.

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40

Donovan, Diane, Mike Grannell, and Emine Şule Yazıcı. "Embedding partial Latin squares in Latin squares with many mutually orthogonal mates." Discrete Mathematics 343, no. 6 (2020): 111835. http://dx.doi.org/10.1016/j.disc.2020.111835.

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41

Brawley, J. V., and Gary L. Mullen. "Infinite Latin squares containing nested sets of mutually orthogonal finite Latin squares." Publicationes Mathematicae Debrecen 39, no. 1-2 (2022): 135–41. http://dx.doi.org/10.5486/pmd.1991.39.1-2.09.

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42

Lewis, James R. "Pairs of Latin Squares to Counterbalance Sequential Effects and Pairing of Conditions and Stimuli." Proceedings of the Human Factors Society Annual Meeting 33, no. 18 (1989): 1223–27. http://dx.doi.org/10.1177/154193128903301812.

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This paper discusses methods with which one can simultaneously counterbalance immediate sequential effects and pairing of conditions and stimuli in a within-subjects design using pairs of Latin squares. Within-subjects (repeated measures) experiments are common in human factors research. The designer of such an experiment must develop a scheme to ensure that the conditions and stimuli are not confounded, or randomly order stimuli and conditions. While randomization ensures balance in the long run, it is possible that a specific random sequence may not be acceptable. An alternative to randomiza

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43

Alpoge, Levent. "Square-root cancellation for the signs of Latin squares." Combinatorica 37, no. 2 (2015): 137–42. http://dx.doi.org/10.1007/s00493-015-3373-7.

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44

CAVENAGH, NICHOLAS J., and IAN M. WANLESS. "THERE ARE ASYMPTOTICALLY THE SAME NUMBER OF LATIN SQUARES OF EACH PARITY." Bulletin of the Australian Mathematical Society 94, no. 2 (2016): 187–94. http://dx.doi.org/10.1017/s0004972716000174.

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A Latin square is reduced if its first row and first column are in natural order. For Latin squares of a particular order$n$, there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order$n\rightarrow \infty$.

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45

Parker, E. T., and Lawrence Somer. "A Partial Generalization of Mann's Theorem Concerning Orthogonal Latin Squares." Canadian Mathematical Bulletin 31, no. 4 (1988): 409–13. http://dx.doi.org/10.4153/cmb-1988-059-7.

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AbstractLetn = 4t+- 2, where the integert ≧ 2. A necessary condition is given for a particular Latin squareLof ordernto have a complete set ofn — 2mutually orthogonal Latin squares, each orthogonal toL.This condition extends constraints due to Mann concerning the existence of a Latin square orthogonal to a given Latin square.

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46

Rathod, Abhay B., and Sanjay M. Gulhane. "An Efficient Parallel Algorithm for Latin Square Design: A Multi Core CPU Approach." International Journal of System Modeling and Simulation 2, no. 2 (2017): 27. http://dx.doi.org/10.24178/ijsms.2017.2.2.27.

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Abstract— The theory of Latin squares is very important tool in design theory. Like much of design theory, Latin squares have various applications in statistics, finite geometries and experimental design, to name a few. In this paper, we proposed an efficient parallel algorithm for Latin square design which have desirable properties for parallel array access. These squares provide conflict free access to various subsets of an n x n array using n memory modules. A transversal of such a square is a set of n entries such that no two entries share the same row, column or symbol. We present a gener

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47

Evangelaras, H., and C. Koukouvinos. "Interaction detection in Latin and Hyper-latin squares." Journal of Discrete Mathematical Sciences and Cryptography 9, no. 2 (2006): 341–48. http://dx.doi.org/10.1080/09720529.2006.10698083.

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48

Shen, Honglian, Xiuling Shan, Ming Xu, and Zihong Tian. "A New Chaotic Image Encryption Algorithm Based on Transversals in a Latin Square." Entropy 24, no. 11 (2022): 1574. http://dx.doi.org/10.3390/e24111574.

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In this paper, a new combinatorial structure is introduced for image encryption, which has an excellent encryption effect on security and efficiency. An n-transversal in a Latin square has the function of classifying all the matrix’s positions, and it can provide a pair of orthogonal Latin squares. Employing an n-transversal of a Latin square, we can permutate all the pixels of an image group by group for the first time, then use two Latin squares for auxiliary diffusion based on a chaotic sequence, and finally, make use of a pair of orthogonal Latin squares to perform the second scrambling. T

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49

De las Cuevas, Gemma, Tim Netzer, and Inga Valentiner-Branth. "Magic squares: Latin, semiclassical, and quantum." Journal of Mathematical Physics 64, no. 2 (2023): 022201. http://dx.doi.org/10.1063/5.0127393.

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Quantum magic squares have recently been introduced as a “magical” combination of quantum measurements. In contrast to quantum measurements, they cannot be purified (i.e., dilated to a quantum permutation matrix)—only the so-called semiclassical ones can. Purifying establishes a relation to an ideal world of fundamental theoretical and practical importance; the opposite of purifying is described by the matrix convex hull. In this paper, we prove that semiclassical magic squares can be purified to quantum Latin squares, which are “magical” combinations of orthonormal bases. Conversely, we prove

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50

Dębski, Michał, and Jarosław Grytczuk. "Latin squares from multiplication tables." Journal of Combinatorial Designs 29, no. 5 (2021): 331–36. http://dx.doi.org/10.1002/jcd.21769.

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