Latin Rectangles with the Constant Row-Column Intersection Property

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. \(r \leq \min\{m,n\}\)

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