Three mutually orthogonal idempotent Latin squares of orders 22 and 26 (Q1918185)

A Latin square based on \(\{1, 2,\dots, n\}\) is called idempotent if its \((i, i)\) entry is \(i\). It is known (see \textit{F. E. Bennett}, \textit{K. T. Phelps}, \textit{C. A. Rodger} and \textit{L. Zhu} [Constructions of perfect Mendelsohn designs, Discrete Math. 103, No. 2, 139-151 (1992; Zbl 0756.05012)]) that there are three mutually orthogonal idempotent Latin squares of order \(n\) for any integer \(n\geq 5\) and \(n\neq 6, 10, 18, 22, 26\). The authors of the present article show that the result is in fact true for \(n= 22\) and 26, and they construct such Latin squares. They can be used to construct a \((110, 5, 1)\)-PMD and a \((130, 5, 1)\)-PMD (for the notation, see Zbl 0851.05019).