Geodesic
$k$-splitted and $k$-homogeneous Latin squares and their transversals
Matematičeskie voprosy kriptografii, Tome 11 (2020) no. 3, pp. 5-19 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the $k$-splitted Latin squares, i.e. Latin squares of order $kn$ with elements from $\left\{ {0, \ldots ,kn - 1} \right\}$ such that after reducing modulo $n$ we obtain $\left( {kn \times kn} \right)$-matrix consisting of $k^2$ Latin squares of order $n$. If these $k^2$ Latin squares of order $n$ are identical, the original Latin square of order $kn$ is called $k$-homogeneous. The precise number of all $k$-homogeneous and lower bound for the number of all $k$-splitted Latin squares are found. Some characteristics of transversals for $k$-splitted Latin squares are described. 
@article{MVK_2020_11_3_a0,
     author = {V. V. Borisenko},
     title = {$k$-splitted and $k$-homogeneous {Latin} squares and their transversals},
     journal = {Matemati\v{c}eskie voprosy kriptografii},
     pages = {5--19},
     year = {2020},
     volume = {11},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MVK_2020_11_3_a0/}
}
TY  - JOUR
AU  - V. V. Borisenko
TI  - $k$-splitted and $k$-homogeneous Latin squares and their transversals
JO  - Matematičeskie voprosy kriptografii
PY  - 2020
SP  - 5
EP  - 19
VL  - 11
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/MVK_2020_11_3_a0/
LA  - ru
ID  - MVK_2020_11_3_a0
ER  - 
%0 Journal Article
%A V. V. Borisenko
%T $k$-splitted and $k$-homogeneous Latin squares and their transversals
%J Matematičeskie voprosy kriptografii
%D 2020
%P 5-19
%V 11
%N 3
%U http://geodesic.mathdoc.fr/item/MVK_2020_11_3_a0/
%G ru
%F MVK_2020_11_3_a0
V. V. Borisenko. $k$-splitted and $k$-homogeneous Latin squares and their transversals. Matematičeskie voprosy kriptografii, Tome 11 (2020) no. 3, pp. 5-19. http://geodesic.mathdoc.fr/item/MVK_2020_11_3_a0/

[1] Sachkov V. N., Vvedenie v kombinatornye metody diskretnoi matematiki, MTsNMO, M., 2004, 421 pp.

[2] Sachkov V. N., Kombinatornye metody diskretnoi matematiki, Nauka, M., 1977, 320 pp.

[3] Sachkov V. N., Kurs kombinatornogo analiza, NITs В«Regulyarnaya i khaoticheskaya dinamikaВ», M.–Izhevsk, 2013, 336 pp.

[4] Glukhov M. M., Elizarov V. P., Nechaev A. A., Algebra, Lan, M., 2015, 608 pp.

[5] Borisenko V. V., “O transversalyakh raspavshikhsya latinskikh kvadratov chetnogo poryadka”, Matematicheskie voprosy kriptografii, 5:1 (2014), 5–25

[6] Borisenko V. V., “O transversalyakh odnorodnykh latinskikh kvadratov”, Matematicheskie voprosy kriptografii, 6:3 (2015), 5–17 | MR

[7] Borisenko V. V., “O transversalyakh raspavshikhsya latinskikh kvadratov s tozhdestvennoi podstanovkoi ${\chi _{ABCD}}$”, Matematicheskie voprosy kriptografii, 11:1 (2020), 9–26 | MR

[8] Borisenko V. V., “Neprivodimye $n$-kvazigruppy sostavnogo poryadka”, Kvazigruppy i lupy, Matematicheskie issledovaniya, 51, В«ShtiintsaВ», Kishinev, 1979, 38–42

[9] C. J. Colbourn, J. H. Dinitz (eds.), Handbook of combinatorial designs, 2nd ed., Chapman/CRC Press, 2006, 990 pp. | MR

[10] Wanless I. M., “A generalization of transversals for Latin squares”, Electron. J. Comb., 9:R12 (2002) | MR | Zbl

[11] Wanless I. M., “Transversals in Latin squares”, Quasigroups and related systems, 15, Institute of Mathematics Academy of Sciences Moldova, 2007, 169–190 | MR

[12] Wanless I. M., “Transversals in Latin squares: a survey”, Surveys in Combinatorics, London Math. Soc. Lect. Note Ser., 392, Cambr. Univ. Press, 2011, 403–437 | MR | Zbl

[13] McKay B. D., Meynert A., Myrvold W. J., “Small Latin squares, quasigroups and loops”, J. Combin. Des., 15:2 (2007), 98–119 | DOI | MR | Zbl

[14] Brown J. W., Parker E. T., “More on order 10 turn-squares”, Ars Combinatoria, 35 (1993), 125–127 | MR | Zbl