Investigating the Existence of Costas Latin Squares via Satisfiability Testing.

Costas Latin squares are important combinatorial structures in combinatorial design theory. Some Costas Latin squares are found in recent years, but there are still some open problems about the existence of Costas Latin squares with specified properties including idempotency, orthogonality, and certain quasigroup properties. In this paper, we describe an efficient method for solving these problems using state-of-the-art SAT solvers. We present new results of Costas Latin squares with specified properties of even order \(n \le 10\). It is found that within this order range, most Costas Latin squares with such properties don’t exist except for a few cases. The non-existence can be certified since SAT solvers can produce a formal proof. Experimental results demonstrate the effectiveness of our method.