In this paper we propose an algorithm for enumerating diagonal Latin squares of small order. It relies on specific properties of diagonal Latin squares to employ symmetry breaking techniques, and on several heuristic optimizations and bit arithmetic techniques to make use of computational power of stateof-the-art CPUs. Using this approach we enumerated diagonal Latin squares of order at most 9,...

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 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. Transversals are closely related to the notions of complete mappings and orthomorphisms in (quasi)groups, and are fundamental to the concept of mutually orthogon...

We introduce quantum Latin squares, combinatorial quantum objects which generalize classical Latin squares. We show that quantum Latin squares can be seen as weakened versions of mutually-unbiased bases (MUBs). Our main results use quantum Latin squares to give a new construction of unitary error bases (UEBs), basic structures in quantum information which lie at the heart of procedures such as ...

It is well known that if n is even, the addition table for the integersmodulo n (whichwe denote by Bn) possesses no transversals.We show that ifn is odd, then the number of transversals in Bn is at least exponential in n. Equivalently, for odd n, the number of diagonally cyclic latin squares of order n, the number of completemappings or orthomorphisms of the cyclic group of order n, the number ...