In this paper we consider four basic multidimensional matrix operations (outer product, Kronecker product, contraction, and projection) and two derivative operations (dot and circle products). We start with the interrelations between these operations and deduce some of their algebraic properties. Next, we study their action on k-stochastic matrices. At last, we prove several relations on the permanents of products of multidimensional matrices. In particular, we obtain that the permanent of the dot product of nonnegative multidimensional matrices is not less than the product of their permanents and show that inequalities on the Kronecker product of nonnegative 2-dimensional matrices cannot be extended to the multidimensional case.

The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. A nonnegative matrix whose every 1‐dimensional plane sums to 1 is called polystochastic. A latin square of order n is an n×n array of n symbols in which each symbol occurs exactly once in each row and each column. A transversal of such a square is a set of n entries such that no two entries share the same row, column, or symbol. Let T(n) be the maximum number of transversals over all latin squares of order n. Here, we prove that over the set of multidimensional polystochastic matrices of order n the permanent has a local extremum at the uniform matrix for whose every entry is equal to 1/n. Also, we obtain an asymptotic value of the maximal permanent for a certain set of nonnegative multidimensional matrices. In particular, we get that the maximal permanent of polystochastic matrices is asymptotically equal to the permanent of the uniform matrix, whence as a corollary we have an upper bound on the number of transversals in latin squares T(n)≤nne−2n+o(n).

We consider the problem of existence of perfect 2‐colorings (equitable 2‐partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we observe known constructions of perfect colorings and propose some new ones giving new parameters. At last, we deduce which parameters of colorings are covered by these constructions and give tables of admissible parameters of 2‐colorings in Hamming graphs H ( n , q ) for small n and q. Using the connection with perfect colorings, we construct an orthogonal array OA(2048, 7, 4, 5).

A d-ary quasigroup of order n is a d-ary operation over a set of cardinality n such that the Cayley table of the operation is a d-dimensional latin hypercube of the same order. Given a binary quasigroup G, the d-iterated quasigroup G [d] is a d-ary quasigroup that is a d-time composition of G with itself. A k-multiplex (a k-plex) K in a d-dimensional latin hypercube Q of order n or in the corresponding d-ary quasigroup is a multiset (a set) of kn entries such that each hyperplane and each symbol of Q is covered by exactly k elements of K. A transversal is a 1-plex.In this paper we prove that there exists a constant c(G, k) such that if a d-iterated quasigroup G of order n has a k-multiplex then for large d the number of its k-multiplexes is asymptotically equal to c(G, k) (kn)! k! n d−1

An n-ary quasigroup f of order q is an n-ary operation over a set of cardinality q such that the Cayley table of the operation is an n-dimensional latin hypercube of order q. A transversal in a quasigroup f (or in the corresponding latin hypercube) is a collection of q (n + 1)-tuples from the Cayley table of f , each pair of tuples differing at each position. The problem of transversals in latin hypercubes was posed by Wanless in 2011.An n-ary quasigroup f is called reducible if it can be obtained as a composition of two quasigroups whose arity is at least 2, and it is completely reducible if it can be decomposed into binary quasigroups.In this paper we investigate transversals in reducible quasigroups and in quasigroups of order 4. We find a lower bound on the number of transversals for a vast class of completely reducible quasigroups. Next we prove that, except for the iterated group Z 4 of even arity, every n-ary quasigroup of order 4 has a transversal. Also we obtain a lower bound on the number of transversals in quasigroups of order 4 and odd arity and count transversals in the iterated group Z 4 of odd arity and in the iterated group Z 2 2 . All results of this paper can be regarded as those concerning latin hypercubes.