Latin squares from multiplication tables

Author(s):  
Michał Dębski ◽  
Jarosław Grytczuk
Author(s):  
P. A. Kadiev ◽  
I. P. Kadiev

Objective. Formation of structurally perfect orthogonal Latin squares by the method of index ordering of the multiplication table elements of n-sets based on the multiplication table. Methods. Orthogonal Latin squares are formed by the method of index structuring of n-set multiplication tables. Results. A method is proposed for constructing structurally perfect orthogonal Latin squares of pairs of indexed finite sets of odd dimension, based on the index ordering of an nxn-array of elements in the multiplication table. A distinctive feature of the proposed method for constructing structurally perfect orthogonal squares from elements of two indexed sets of the same dimension is the use by the authors of the method of permutations of elements of the original nxn-matrix configurations, with the formation of index-ordered or index-structured combinatorial configurations. Conclusion. The use of the method for constructing a family of orthogonal Latin squares for pairs of indexed finite sets of the same odd dimension by the elements forming their multiplication table by the method of index structuring based on the principle of functional dependency of the index values on pairs of set elements and index values on pairs of elements from its environment allows creating a specific class orthogonal configuration, which, in terms of element indices, easily demonstrates their orthogonality.


Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 285
Author(s):  
Laura M. Johnson ◽  
Stephanie Perkins

This communication provides a discussion of a scheme originally proposed by Falcón in a paper entitled “Latin squares associated to principal autotopisms of long cycles. Applications in cryptography”. Falcón outlines the protocol for a cryptographical scheme that uses the F-critical sets associated with a particular Latin square to generate access levels for participants of the scheme. Accompanying the scheme is an example, which applies the protocol to a particular Latin square of order six. Exploration of the example itself, revealed some interesting observations about both the structure of the Latin square itself and the autotopisms associated with the Latin square. These observations give rise to necessary conditions for the generation of the F-critical sets associated with certain autotopisms of the given Latin square. The communication culminates with a table which outlines the various access levels for the given Latin square in accordance with the scheme detailed by Falcón.


2015 ◽  
Vol 32 (2) ◽  
pp. 543-552 ◽  
Author(s):  
Nicholas J. Cavenagh ◽  
Vaipuna Raass
Keyword(s):  

2006 ◽  
Vol 90 (519) ◽  
pp. 478-481 ◽  
Author(s):  
Emanuel Emanouilidis
Keyword(s):  

2011 ◽  
Vol 28 (2) ◽  
pp. 30-39 ◽  
Author(s):  
Seung Eun Lee ◽  
Yoon Seok Yang ◽  
G S Choi ◽  
Wei Wu ◽  
R Iyer

2012 ◽  
Vol 20 (6) ◽  
pp. 265-277 ◽  
Author(s):  
Leonard H. Soicher
Keyword(s):  

2009 ◽  
Vol 107 (4) ◽  
pp. 1165-1171 ◽  
Author(s):  
Andrew W. Subudhi ◽  
Ronney B. Panerai ◽  
Robert C. Roach

We investigated the effect of acute hypoxia (AH) on dynamic cerebral autoregulation (CA) using two independent assessment techniques to clarify previous, conflicting reports. Twelve healthy volunteers (6 men, 6 women) performed six classic leg cuff tests, three breathing normoxic (FiO2 = 0.21) and three breathing hypoxic (FiO2 = 0.12) gas, using a single blinded, Latin squares design with 5-min washout between trials. Continuous measurements of middle cerebral artery blood flow velocity (CBFv; DWL MultiDop X2) and radial artery blood pressure (ABP; Colin 7000) were recorded in the supine position during a single experimental session. Autoregulation index (ARI) scores were calculated using the model of Tiecks et al. (Tiecks FP, Lam AM, Aaslid R, Newell DW. Stroke 26: 1014–1019, 1995) from ABP and CBFv changes following rapid cuff deflation (cuff ARI) and from ABP to CBFv transfer function, impulse, and step responses (TFA ARI) obtained during a 4-min period prior to cuff inflation. A new measure of %CBFv recovery 4 s after peak impulse was also derived from TFA. AH reduced cuff ARI (5.65 ± 0.70 to 5.01 ± 0.96, P = 0.04), TFA ARI (4.37 ± 0.76 to 3.73 ± 0.71, P = 0.04), and %Recovery (62.2 ± 10.9% to 50.8 ± 9.9%, P = 0.03). Slight differences between TFA and cuff ARI values may be attributed to heightened sympathetic activity during cuff tests as well as differential sensitivity to low- and high-frequency components of CA. Together, results provide consistent evidence that CA is impaired with AH. In addition, these findings demonstrate the potential utility of TFA ARI and %Recovery scores for future CA investigations.


2008 ◽  
Vol 16 (6) ◽  
pp. 477-492 ◽  
Author(s):  
Judith Egan ◽  
Ian M. Wanless
Keyword(s):  

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