N(n) and ν(n): Similarities and Differences

Much has been written about the construction of sets of mutually orthogonal latin squares (MOLS). In [8], a lengthy survey of these constructions is given. Existence of MOLS is tabulated in [1], historical information appears in [9, 21], and proofs of many of the existence results appear in [1, 4, 21]. Rather than repeat these surveys here, we instead explore how some of the available constructions can be applied to a similar problem that has not been studied quite as extensively. We begin with some definitions. A latin square of side n is an n × n array in which each cell contains a single element from an n-set S, such that each element occurs exactly once in each row and exactly once in each column. A latin square L of side n is symmetric if L(i, j) = L(j, i) for all 1 ≤ i, j ≤ n. A latin square L of side n is idempotent if L(i, i) = i for all 1 ≤ i ≤ n. Two latin squares L and L′ of the same order are orthogonal if L(a, b) = L(c, d) and L′(a, b) = L′(c, d), implies a = c and b = d. An equivalent definition for orthogonality is: Two latin