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 of which are 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 [1], (Theorem 2C). Hence they state: "We do not know whether we can get 11 or 12 constraints in any other way." This paper shows first that a 10-rowed orthogonal array, constructed by the geometrical method, cannot be extended to an 11-rowed orthogonal array. It then shows that the number of constraints does not exceed 11. The problem of construction of an orthogonal array with 11 constraints remains unsolved. This summary should serve, as well, as a correction to the statement made in the abstract, "A remark on the geometrical method of construction of an orthogonal array," published in the Annals of Mathematical Statistics, Vol. 25 (1954), p. 177-178, which claimed the nonexistence of an orthogonal array of 11 constraints also.