A direct proof of the non-existence of a MOL(6)
It is well known that a Latin Square of order 6 has no orthogonal mate. Many proofs are known, some of which are very short, see for instance (Betten in Unterricht 36:449–453, 1983; Beth et al. in Design theory, Bibliographisches Institut Mannheim Wien, Zürich, 1985; Tarry in Comptes Rendus Ass Franc Sci Nat 1900(2), 170–203, 1901; Stinson in J Comb Theory A 36:373–376, 1984). This paper provides a short proof of this fact which avoids a case distinction on the isomorphism types of the Latin Square. We observe that any Latin Square of order 6 falls within exactly one of three categories. Either it has two rows which form a permutation whose cycle type is three transpositions. Or it has no subsquare of order 2, or it is a single Latin Square with symmetry group the rotation group of the cube of order 24. In each case, the nonexistence of an orthogonal mate can be seen quickly.
Preview
Rights
Use and reproduction:
Please note that individual components of the publication may be subject to other licensing or copyright conditions.