Orthogonal Arrays with Parameters OA(s3,s2+s+1,s,2) and 3-Dimensional Projective Geometries

There are many nonisomorphic orthogonal arrays with parameters OA(s3, s2 + s+1, s, 2) although the existence of the arrays yields many restrictions. We denote this by OA(3, s) for simplicity. V.D. Tonchev showed that for even the case of s = 3, there are at least 68 nonisomorphic orthogonal arrays. The arrays that are constructed by the n−dimensional finite spaces have parameters OA(s, (s − 1)/(s− 1), s, 2). They are called Rao-Hamming type. In this paper we characterize the OA(3, s) of 3-dimensional Rao-Hamming type. We prove several results for a special type of OA(3, s) that satisfies the following condition: For any three rows in the orthogonal array, there exists at least one column, in which the entries of the three rows equal to each other. We call this property α-type. We prove the following. (1) An OA(3, s) of α-type exists if and only if s is a prime power. (2) OA(3, s)s of α-type are isomorphic to each other as orthogonal arrays. (3) An OA(3, s) of α-type yields PG(3, s). (4) The 3-dimensional Rao-Hamming is an OA(3, s) of α-type. (5) A linear OA(3, s) is of α-type.