Latin Rectangles with the Constant Row-Column Intersection Property

Abstract

Let $L$ be an $n\times m$ Latin rectangle on a set of $v$ symbols with the property that each symbol occurs in precisely $r$ cells of $L$. Then $L$ is said to have the row-column intersection property if each row and column of $L$ have precisely $r$ symbols in common. It is shown here that the trivial necessary conditions

1. $rv=mn$ and
2. $r\le min\{m,n\}$

are sufficient to guarantee the existence of such a Latin rectangle.