OPTIMAL SEMI-LATIN SQUARES WITH SIDE-6 AND BLOCK SIZE-2

An (n x n)/k semi-Latin square is like an n x n Latin square except th at there are k letters in each cell. Each of the nk letters occurs onc e in each row and once in each column. Designs for experiments are ass essed according to the statistical concept of efficiency factor. A hig h efficiency factor corresponds to low variances of within-block estim ators. There are four widely used measures of the efficiency factor of a design: for each, any design which maximizes the value of the effic iency factor among a given class of designs is said to be optimal in t hat class. Previous theory gives optimal semi-Latin squares for variou s values of k for all values of n except for n = 6. In this paper rye therefore examine (6 x 6)/2 semi-Latin squares. We restrict attention to those semi-Latin squares whose quotient block designs are regular-g raph designs, because a plausible and widely believed conjecture is th at optimal regular-graph designs are optimal overall. For each of the four measures of efficiency factor, rye find the optimal (6 x 6)/2 sem i-Latin square among regular-graph semi-Latin squares of that size.