Abstract
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.
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Acknowledgments
This work has been supported by the National Natural Science Foundation of China (NSFC) under grant No.61972384, and the Key Research Program of Frontier Sciences, Chinese Academy of Sciences under grant number QYZDJ-SSW-JSC036. Feifei Ma is also supported by the Youth Innovation Promotion Association CAS under grant No. Y202034. We thank professor Lie Zhu at SooChow university for suggesting these open problems and his valuable advice. We also thank the anonymous reviewers for their comments and suggestions.
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Jin, J., Lv, Y., Ge, C., Ma, F., Zhang, J. (2021). Investigating the Existence of Costas Latin Squares via Satisfiability Testing. In: Li, CM., Manyà, F. (eds) Theory and Applications of Satisfiability Testing – SAT 2021. SAT 2021. Lecture Notes in Computer Science(), vol 12831. Springer, Cham. https://doi.org/10.1007/978-3-030-80223-3_19
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DOI: https://doi.org/10.1007/978-3-030-80223-3_19
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