Square Lattice Designs
The characteristic features of the square lattice designs introduced by Yates (1936) are that the number of treatments is a perfect square and the block size is the square root of this number. If the design has two replications of the treatments, it is called a simple lattice; if it has 3 replications it is called a triple lattice and so on. In general, if the number of replications is i, it is called an i-ple lattice design.
Square lattice designs can be constructed as follows:
Let there be v = s2 treatments, numbered as 1, 2, …, s2. Arrange these treatment numbers in the form of a s ´ s square array in natural order, i.e., in a standard array. The contents of each of the s rows of this array are taken to form blocks giving a set of s blocks; another set of s blocks forming another complete replication is obtained by taking the contents of each of the s columns of this array. It can be check easily that this simple lattice design is a PBIB design with a Latin square association scheme, that is, a L2-PBIB design.
Next, a s ´ s Latin square is taken and is superimposed on the above standard array of treatment numbers. The treatment numbers that fall on a particular symbol of the Latin square are taken to form a block. Thus s blocks corresponding to the s symbols of Latin square can be obtained. Again, another Latin square orthogonal to the previous one is taken and from this square also, another set of s blocks is obtained in the same manner. The process is repeated to get further replications. The process is continued till i-2 (≤ s-1, if s is a prime or power of a prime) mutually orthogonal Latin squares are utilized. When the complete set of (s-1) mutually orthogonal Latin squares (if such a set exists) is utilized, the design becomes a balanced (s+1) lattice. A balanced lattice is a BIB design.