Efficient semi-Latin squares

Consider an n \times n square array in which each small square is divided into k plots. A semi-Latin square is an allocation of nk treatments to the plots of such an array so that each treatment occurs once in each row and once in each column. Several different practical situations are discussed which all lead to this same abstract structure.

There are two reasonable models for data from semi-Latin squares. Under the first, all semi-Latin squares are equally efficient, while under the second there is a wider range of efficiencies. Attention is focused on the problem of finding efficient semi-Latin squares for the second model.

There is a family of semi-Latin squares called Trojan squares, which are known to be optimal, as are certain squares derived from Trojan squares. Unfortunately, these do not exist for al pairs of values of n and k. Recent agricultural experiments have required efficient semi-Latin squares for some of these other values of n and k. New designs for these values are presented and their efficiencies and possible optimality discussed.