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Example text

Am+d exists for which, for each 1 $ Ie < m, the differences represent the m cyclotomic classes of GF( mt + 1) (compute subscripts modulo m + 2 as needed). In other words, for a fixed Ie, if CJi+1c - CJi = ",mil+a and aj+1c - aj = ",my +tI , we find that a "1. f3 (mod m). Then form a single column of length m + 2 whose first entry is empty, and whose remaining entries are (all"" am+t). Form t columns by multiplying this column by the powers of ",m. From each of these t columns, form m + 2 columns by taking the m + 2 cyclic shifts of the column.

Let A be an OA(k,n) on the n symbols in X. On V = X x {I, ... ,k} (a set of size kn), form a set B of k-sets as follows. i' i) : I :$ i :$ k} in. B. Then let g be the partition of V whose classes are {X x {i} : I :$ i:$ k}. Then (V,g,B) is a TD(k,n). (k, n) from a TD(k, n). Thus, k MOLS of side n, a TD(k + 2, n), and an OA(k + 2, n) are all equivalent. In each of these disguises, mutually orthogonal latin squares have been extensively studied and lue central in combinatorial design theory and in experimental design theory.

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