Some New Non-Cyclic Latin Squares That Have Cyclic and Youden Properties

This note gives what is believed to be the first published example of a symmetric \(11 \times 11\) Latin square which, although not cyclic, has the property that the permutation between any two rows is an \(11\)-cycle. The square has the further property that two subsets of its rows constitute \(5 \times 11\) Youden squares. The note shows how this \(11 \times 11\) Latin square can be obtained by a general construction for \(n \times n\) Latin squares where \(n\) is prime with \(n \geq 11\). The permutation between any two rows of any Latin square obtained by the general construction is an \(n\)-cycle; two subsets of \((n-1)/2\) rows from the Latin square constitute Youden squares if \(n \equiv 3 \pmod{8}\).