Further Remark on the Maximum Number of Constraints of an Orthogonal Array

Abstract

R. C. Bose and K. A. Bush [1] showed how to make use of the maximum number of points, no three collinear, in finite projective spaces for the construction of orthogonal arrays. In particular, this enabled them to construct an orthogonal array (81, 10, 3, 3). They proved, on the other hand, that in the case considered the maximum number of constraints does not exceed 12. Hence they state, "We do not know whether we can get 11 or 12 constraints in any other way." A partial solution to this problem was given by the author [2]. It was shown that the number of constraints cannot exceed 11. The purpose of this paper is to give a complete solution to the above stated problem, namely, to prove that no way exists which could give a number of constraints, of the considered orthogonal array, greater than ten. As a consequence of the proof it follows also that any orthogonal array with ten constraints satisfies a unique algebraic solution. It is not known, however, whether the arrays constructed by the geometrical method form the totality of orthogonal arrays of the considered type.