Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
latin squares; bachelor squares; monogamous squares; prolongation
latin squares; bachelor squares; monogamous squares; prolongation
Summary:
We consider two classes of latin squares that are prolongations of Cayley tables of finite abelian groups. We will show that all squares in the first of these classes are confirmed bachelor squares, squares that have no orthogonal mate and contain at least one cell though which no transversal passes, while none of the squares in the second class can be included in any set of three mutually orthogonal latin squares.
References:
[1] Belyavskaya G.B.: Generalized extension of quasigroups. (Russian), Mat. Issled. 5 (1970), 28–48. MR 0284533
[2] Belyavskaya G.B.: Contraction of quasigroups. I. (Russian), Bul. Akad. Stiince RSS Moldoven (1970), 6–12. MR 0279219
[3] Belyavskaya G.B.: Contraction of quasigroups. II. (Russian), Bul. Akad. Stiince RSS Moldoven (1970), 3–17. MR 0284531
[4] Danziger P., Wanless I.M., Webb B.S.: Monogamous latin squares. J. Combin. Theory Ser. A 118 (2011), 796–807. DOI 10.1016/j.jcta.2010.11.011 | MR 2745425 | Zbl 1232.05033
[5] Deriyenko I.I., Dudek W.A.: On prolongation of quasigroups. Quasigroups and Related Systems 16 (2008), 187–198. MR 2494876
[6] Evans A.B.: Latin squares without orthogonal mates. Des. Codes Crypt. 40 (2006), 121–130. DOI 10.1007/s10623-006-8153-3 | MR 2226287 | Zbl 1180.05022
[7] Paige L.J.: A note on finite abelian groups. Bull. Amer. Math. Soc. 53 (1947), 590–593. DOI 10.1090/S0002-9904-1947-08842-X | MR 0020990 | Zbl 0033.15101
[8] Wanless I.M.: Transversals in latin squares: a survey. Surveys in combinatorics 2011, pp. 403–437, London Math. Soc. Lecture Note Ser., 392, Cambridge Univ. Press, Cambridge, 2011. MR 2866738 | Zbl 1226.05067
[9] Wanless I.M., Webb B.S.: The existence of latin squares without orthogonal mates. Des. Codes Cryptogr. 40 (2006), 131–135. DOI 10.1007/s10623-006-8168-9 | MR 2226288 | Zbl 1180.05023
Search
Partner of
