On computational search for quasi-orthogonal systems of Latin squares which are close to orthogonal systems.

Evgenia Beley, Alexandr Semenov


In the paper we study the applicability of modern algorithms of computational logic to problems of finding some combinatorial designs. In particular, we consider the well-known open problem – to answer whether there exist three mutually orthogonal Latin squares of order 10. We propose an iterative procedure to search for such a triple. At each iteration, it constructs the so-called quasi–orthogonal systems. We use the orthogonality index metric that makes it possible to measure how close is the quasi–orthogonal system to the orthogonal one. To construct quasi-orthogonal systems with specified orthogonality index we employ the state-of-the-art algorithms for solving Boolean satisfiability problem (SAT). The results of our computational experiments show that the SAT-solvers can successfully be used to search for new combinatorial designs based on Latin squares.

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