Perpendicular Rectangular Latin Arrays

A set { A 1 , A 2 ,..., A t } of rectangular arrays, each defined on a symbol set X , is said to be t -perpendicular if each t -element subset of X occurs precisely once when the arrays are superimposed. We investigate the existence of sets of r by s rectangular arrays which are row-Latin, column-Latin and t -perpendicular. For example, we show that for all odd n , there exists a pair of row- and column-Latin 2-perpendicular r by s arrays with symbol set X of size n if and only if and r , s ≤ n .