MINISTRY OF SCIENCE AND EDUCATION OF THE REPUBLIC OF AZERBAIJAN

# CENTRAL SYMMETRY PROPERTIES FOR DIAGONAL LATIN SQUARES

### Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, Vitaly S. Titov

Latin squares and diagonal Latin squares are well known combinatorial objects connected with specific open mathematical problems. Most famous of them is the problem of existing triple of mutually orthogonal (diagonal) Latin squares of order 10 that are not proven or disproven at this moment. Best approximations of it is connected with symmetries (automorphisms) under Latin squares. Pseudo triple of diagonal Latin squares with world record 274 orthogonality characteristics is based on square with horizontal (plane) symmetry and it is shown that these characteristics can’t be improved with this type of symmetry. In this work, we introduce a definition of different type of symmetry – the central symmetry for diagonal Latin squares and show the corresponding mathematical relations between the cells of the Latin square. Moreover, we describe the properties of the new symmetry and perform enumeration of corresponding squares of order 9 at most (pp.3-8).

Keywords: diagonal Latin squares, symmetries, orthogonal mates, pseudo triples, orthogonality characteristics.
DOI : 10.25045/jpit.v10.i2.01
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