Optimal semi-Latin squares with side six and block size two

An (n \times n)/k semi-Latin square is like an n \times n Latin square except that there are k letters in each cell. Each of the nk letters occurs once in each row and once in each column.

Designs for experiments are assessed according to the statistical concept of efficiency factor. A high efficiency factor corresponds to low variances of within-block estimators. There are four widely used measures of the efficiency factor of a design: for each, any design which maximizes the value of the efficiency factor among a given class of designs is said to be optimal in that class.

Previous theory gives optimal semi-Latin squares for various values of k for all values of n except for n=6. In this paper we therefore examine (6 \times 6)/2 semi-Latin squares. We restrict attention to those semi-Latin squares whose quotient block designs are regular-graph designs, because a plausible and widely believed conjecture is that optimal regular-graph designs are optimal overall.

For each of the four measures of efficiency factor, we find the optimal (6\times 6)/2 semi-Latin square among regular-graph semi-Latin squares of that size.