2016/12/06 by Anna A. Taranenko, Taranenko, Anna, Anna Taranenko
Engineering · Mathematics · #graph theory and CDMA systems #Mathematics and Applications #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.1612.01797
An n-ary quasigroup f of order q is an n-ary operation over a set of\ncardinality q such that the Cayley table of the operation is an\nn-dimensional latin hypercube of order q. A transversal in a quasigroup f\n(or in the corresponding latin hypercube) is a collection of q (n+1)-tuples\nfrom the Cayley table of f, each pair of tuples differing at each position.\nThe problem of transversals in latin hypercubes was posed by Wanless in 2011.\n An n-ary quasigroup f is called reducible if it can be obtained as a\ncomposition of two quasigroups whose arity is at least 2, and it is completely\nreducible if it can be decomposed into binary quasigroups.\n In this paper we investigate transversals in reducible quasigroups and in\nquasigroups of order 4. We find a lower bound on the number of transversals for\na vast class of completely reducible quasigroups. Next we prove that, except\nfor the iterated group \ℤ4 of even arity, every n-ary quasigroup\nof order 4 has a transversal. Also we obtain a lower bound on the number of\ntransversals in quasigroups of order 4 and odd arity and count transversals in\nthe iterated group \ℤ4 of odd arity and in the iterated group\n\ℤ22.\n All results of this paper can be regarded as those concerning latin\nhypercubes.\n