Geodesic
On a method of construction of orthogonal quasigroup systems by means of groups
Matematičeskie voprosy kriptografii, Tome 2 (2011), pp. 5-24.

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We suggest a method of construction of orthogonal quasigroups systems and orthogonal Latin squares by means of Frobenius groups. It is proved that this method generalizes many (but not all) existing methods of orthogonal Latin squares construction by means of groups. 
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M. M. Gluhov. On a method of construction of orthogonal quasigroup systems by means of groups. Matematičeskie voprosy kriptografii, Tome 2 (2011), pp. 5-24. http://geodesic.mathdoc.fr/item/MVK_2011_2_a0/

[1] Belousov V. D., Osnovy teorii kvazigrupp i lup, Nauka, M., 1967 | MR | Zbl

[2] Belousov V. D., Algebraicheskie seti i kvazigruppy, Shtiintsa, Kishinev, 1971

[3] Belousov V. D., “Sistemy ortogonalnykh operatsii”, Matem. sb., 77(119):1 (1968), 38–58 | MR | Zbl

[4] Glukhov M. M., Larin S. V., “Abelevy i metabelevy gruppy s regulyarnymi avtomorfizmami i poluavtomorfizmami”, Matem. zametki, 12:6 (1972), 727–738 | Zbl

[5] Kargopolov M. I., Merzlyakov Yu. I., Osnovy teorii grupp, Nauka, M., 1972 | MR

[6] Sachkov V. N., Kombinatornye metody diskretnoi matematiki, Nauka, M., 1977

[7] Kholl M., Kombinatorika, Mir, M., 1970 | MR

[8] Belyavskaya G. B., “Quasigroup power sets and cyclic $S$-systems”, Quasigroups and related Systems, 9 (2002), 1–17 | MR | Zbl

[9] Bose R. C., “On application of the properties of Galois fields to the problem of construction of hyper-Graeco-Latin squares”, Sankhya, 3 (1938), 323–338

[10] Bose R. C., Nair K. R., “On complete sets of orthogonal Latin squares”, Sankhya, 5 (1941), 361–382 | MR | Zbl

[11] Bose R. C., Shrikhande S. S., Parker E. T., “Further results on the constructions of mutually orthogonal Latin squares and the falsity Euler`s conjecture”, Canad. J. Math., 12 (1960), 189–203 | DOI | MR | Zbl

[12] Brack R. H., “Finite nets, I”, Canad. J. Math., 3 (1951), 94–107 | DOI | MR

[13] Brack R. H., A survey of binary systems, Springer-Verlag, Berlin–Heidelberg–Gottingen, 1958 | MR

[14] Denes J., Keedwell A. D., Latin Squares. New Developments in the Theory and Applications, Nord-Holland Publishing Co., Amsterdam, 1981 | MR

[15] Denes J., Mullen G. L., Suchower S. J., “A note on power sets of Latin squares”, J. Combin. Computing, 16 (1994), 27–31 | MR | Zbl

[16] Denes J., “When is there a Latin power set?”, Amer. Math. Monthly, 104 (1997), 563–565 | DOI | MR

[17] Denes J., Owens P. J., “Some new power sets not based on groups”, J. Combin. Theory Ser. A, 85 (1999), 69–82 | DOI | MR | Zbl

[18] Denes J., Petroczki P., A digital encrypting communication systems, Hungarian Patent No 201437A, 1990

[19] Gorenstein D., Finite groups, Harper and Row, New York–Evanston–London, 1968 | MR | Zbl

[20] Huppert B., Endliche Gruppen, v. I, Springer-Verlag, Berlin–Heidelberg–New York, 1967 | MR | Zbl

[21] Johnson D. M., Dulmage A. L., Mendelson N. S., “Orthomorphisms of groups and orthogon Latin squares, I”, Canad. J. Math., 13 (1961), 356–372 | DOI | MR | Zbl

[22] Keedwell A. D., “On $R$-sequenceability and $R_h$-sequenceability of groups”, Combinatorics' 81, Ann. Discrete Math., 78, North-Holland, 1983, 535–548 | MR

[23] Koksma K. K., “Lower bound for the order of a partial transversal in a Latin square”, J. Combinatorial Theory, 7 (1969), 94–95 | DOI | MR | Zbl

[24] Laywine C. F., Mullen G. L., Discrete mathematics using Latin squares, John Wiley and Sons, Inc., New York, 1998 | MR

[25] Leakh I. V., On transformations of orthogonal systems of optrations and algebraic nets, Ph. D. thesis, IM AN RM, Kishinev, 1986

[26] MacNeish H. F., “Euler squares”, Ann. Math., 23 (1921), 221–227 | DOI | MR

[27] Mann H. B., “The construction orthogonal Latin squares”, Ann. Math. Statist., 13 (1942), 418–423 | DOI | MR | Zbl

[28] Norton D. A., “Group of orthogonal row-latin squares”, Pacific J. Math., 2 (1952), 335–341 | MR | Zbl

[29] Paige L. J., “Complete mappings of finite groups”, Pacific J. Math., 1 (1951), 111–116 | MR | Zbl

[30] Paige L. J., Hall M., “Complete mappings of finite groups”, Pacific J. Math., 5 (1955), 541–549 | MR | Zbl

[31] Parker E. T., “Construction of some sets of pairwise orthogonal Latin squares”, Proc. Amer. Math. Soc., 10 (1959), 949–951 | DOI | MR

[32] Parker E. T., “Orthogonal Latin squares”, Proc. Natural Academ. Sci. USA, 45 (1959), 859–962 | DOI | MR

[33] Stevens W., “The Completely orthogonalized Latin squares”, Ann. Eugen., 9 (1939), 269–307 | DOI | MR

[34] Wanless I. M., “Transversals in Latin squares”, Quasigroups and related systems, 15:1 (2007), 169–190 | MR | Zbl