Formation of orthogonal latin squares by index structuring of n-set multiplication tables
https://doi.org/10.21822/2073-6185-2020-47-3-71-81
Abstract
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.
About the Authors
P. A. Kadiev
Daghestan State Technical University
Russian Federation
Russian Federation
Pashay A. Kadiev - Cand. Sci. (Technical), Assoc. Prof., Department of Management and Informatics in Technical Systems and Computer Engineering.
70 I. Shamil Ave., Makhachkala 367026.
I. P. Kadiev
Daghestan State Technical University
Russian Federation
Russian Federation
Islamudin P. Kadiev - Applicant, Department of Management and Informatics in Technical Systems and Computer Engineering.
70 I. Shamil Ave., Makhachkala 367026.
References
1. Vilenkin N.YA. Kombinatorika. - M.: «Nauka»,1969g.,328s. [Vilenkin N. Ya. Combinatorics. - M .: "Science", 1969, 328s. (In Russ)]
2. Tarakanov V.Ye., Aygner M. A. Kombinatornaya teoriya.- M.: Mir,1982, 362s. [Tarakanov V.E., Aigner M.A., Combinatorial theory, Moscow: Mir, 1982, 362p. (In Russ)]
3. Kholl M. Kombinatorika. / Perevod s angliyskogo S.A. Shirokova pod red. A.O. Gel'fanda A.O.i Tarakanova V.Ye //.- M.: Mir,i970.[Hall M. Combinatorics. / Translated from English by S.A. Shirokova, ed. A.O. Gelfand A.O. and Tarakanova V.E. // M .: Mir, 1970 (In Russ)]
4. Stenli R. Perechislitel'naya kombinatorika. M.: Mir, i990. [Stanley R. Enumeration combinatorics - M .: Mir, 1990. (In Russ)]
5. Rybnikov K.A. Vvedeniye v kombinatornyy analiz.- M.: izd. MGU, 1994. [Rybnikov K.A. Introduction to combinatorial analysis. Moscow: ed. Moscow State University, 1994. (In Russ)]
6. http://www.google/ru. Index sorting algorithms for data arrays.
7. Leont'yev V.K. Izbrannyye zadachi kombinatornogo analiza. - M.: izd-vo MGTU im. N.E. Baumana, 200i [Leontiev V.K. Selected problems of combinatorial analysis. - M .: publishing house of MSTU im. N.E. Bauman, 2001 (In Russ)]
8. Volkova V.N. Teoriya sistem i sistemnyy analiz. Uchebnik / V. N. Volkova, A. A. Denisov //- M.: Yurayt, 2015.-615s. [Volkova V.N. Systems theory and systems analysis. Textbook / V. N. Volkova, A. A. Denisov // - M .: Yurayt, 2015.-615s. (In Russ)]
9. Denes J. H., Keedwell A. D. Latin squares: New developments in the theory and applications. Annals of Discrete Mathematics vol. 46. Academic Press. Amsterdam. 1991.
10. Rybnikov K. A. Kombinatornyy analiz. Ocherki istorii. Tekst//— M.: Izd. Mekhmata MGU, 1996. — 124 s. [Rybnikov KA Combinatorial analysis. Essays on history. Text // - M .: Ed. Mehmat Moscow State University, 1996. 124 p. (In Russ)]
11. Andersen J.A. Discrete mathematics and combinatorics. Text // lane. from English - M .: Williams, 2003.
12. Kadiyev P.A., Kadiyev I.P. Algoritmy preobrazovaniya «klassicheskikh» matrits v 2-kh indeksnyye latinskiye kvadraty. Tekst. / P.A Kadiyev, I.P. Kadiyev, M.Z. Zeynalov. // Vestnik Dag. Gos. Tekh. Un-ta, Ti7. 20i0. s.93-99 [Kadiev P.A., Kadiev I.P. Algorithms for transforming "classical" matrices into 2-index Latin squares. Text. / P.A. Kadiev, I.P. Kadiev, M.Z. Zeynalov//Herald of Daghestan State Technical University. Technical Sciences. Vol. 17. 2010. рр.93-99 (In Russ)]
13. Kadiyev P.A. Programma preobrazovaniya matrits metodom latinskikh kvadratov. Tekst. / P.A. Kadiyev, M.Z. Zeynalov. // Svidetel'stvo o gosudarstvennoy registratsii programm dlya EVM №2009616143 ot 09.11.2009g. [Kadiev P.A. A program for converting matrices by the method of Latin squares. Text. / P.A. Kadiev, M.Z. Zeynalov. // Certificate of state registration of computer programs No. 2009616143 dated 09.11.2009. (In Russ)]
14. Kadiyev I.P. Rasseivaniye elementov «paketov oshibok» v informatsionnom massive metodom indeksnoy strukturizatsii. Tekst / P.A Kadiyev, I.P. Kadiyev, Kudayev R.B. // Vestnik DGTU. Tekhnicheskiye nauki. Tom 46. №4, 2019.- s.8i-89 [Kadiev I.P. Scattering of "error packets" elements in the information array by the method of index structuring. Text / P.A. Kadiev, I.P. Kadiev, Kudaev R.B. // Herald of Daghestan State Technical University. Technical Sciences. Vol. 46. No4, 2019. pp.81-89 (In Russ)]
15. Kadiyev I.P. Sistema indeksnoy strukturizatsii kombinatornykh konfiguratsiy metodom rekurrentnykh funktsion-al'nykh sootnosheniy dlya zashchity peredavayemykh po kanalam svyazi dannykh. Tekst./ Kadiyev I.P., Melekhin V.B. // zh. Pribory i sistemy. Upravleniye, kontrol', diagnostika. № 2 ,2019. s.37- 43 [Kadiev I.P. The system of index structuring of combinatorial configurations by the method of recurrent functional relations for the protection of data transmitted over communication channels. Text. / Kadiev I.P., Melekhin V.B. // J. Devices and systems. Management, control, diagnostics. No. 2, 2019 pp. 37-43 (In Russ)]
16. Laywine C. F. and Mullen G. L. Discrete mathematics using Latin squares. New York: Wiley, 1998.
17. Chum C.S. and Zhang X. The Latin squares and the secret sharing schemes // Groups Complex. Cryptol. 2010. Vol. 2. pp. 175-202.
18. Laywine C. F. and Mullen G. L. Discrete mathematics using Latin squares. New York: Wiley, 1998.
19. Glukhov M. M. O primeneniyakh kvazigrupp v kriptografii .Stat'ya. // Prikladnaya diskretnaya matematika. 2008. №2(2). s. 28-32. [Glukhov MM On applications of quasigroups in cryptography. Article. // Applied discrete mathematics. 2008. No. 2 (2). pp. 28-32. (In Russ)]
20. Malykh A.Ye Ob istoricheskom protsesse razvitiya teorii latinskikh kvadratov i nekotorykh ikh prilozheniyakh. Tekst. /. Malykh A.Ye., Danilova V. I. // Vestnik Permskogo universiteta. 2010. Vol. 4(4). S. 95-104. [Malykh AE On the historical development of the theory of Latin squares and some of their applications. Text. /. Malykh A.E., Danilova V.I. // Bulletin of Perm University. 2010. Issue. 4 (4). pp. 95-104. (In Russ)]
21. Trishin A.Ye. Sposob postroyeniya ortogonal'nykh latinskikh kvadratov na osnove podstanovochnykh dvuchlenov konechnykh poley . Tekst.// M.: TVP. [Trishin A.E. A method for constructing orthogonal Latin squares based on substitutional binomials of finite fields. Text // M .: TVP. (In Russ)]
22. Tuzhilin M. E. Ob istorii issledovaniy latinskikh kvadratov Tekst. // Obozreniye prikladnoy i promyshlennoy matematiki. 2012. Tom 19, vypusk 2. S. 226—227. [Tuzhilin M. E. On the history of studies of Latin squares Text.// Review of Applied and Industrial Mathematics. 2012. Vol. i9, Issue 2. рр. 226-227. (In Russ)]
Review
For citations:
Kadiev P.A., Kadiev I.P. Formation of orthogonal latin squares by index structuring of n-set multiplication tables. Herald of Dagestan State Technical University. Technical Sciences. 2020;47(3):71-81. (In Russ.) https://doi.org/10.21822/2073-6185-2020-47-3-71-81
Views:
639
Email this article 
