Orthogonal Arrays of Index Unity

In this paper we shall proceed to generalize the notion of a set of orthogonal Latin squares, and we term this extension an orthogonal array of index unity. In Section 2 we secure bounds for the number of constraints which are the counterpart of the familiar theorem which states that the number of mutually orthogonal Latin squares of side s is bounded above by $s - 1$. Curiously, our bound depends upon whether s is odd or even. In Section 3 we give a method of constructing these arrays by considering a class of polynomials with coefficients in the finite Galois field $GF(s)$, where s is a prime or a power of a prime. In the concluding section we give a brief discussion of designs based on these arrays.