Orthogonal arrays with parameters OA and 3-dimensional projective geometries (Q540058)

Summary: There are many nonisomorphic orthogonal arrays with parameters \(OA(s^3, s^2+ s+ 1,s,2)\) 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 \(OA(s^n,(s^n- 1)1(s- 1), s,2)\). 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 \(\alpha\)-type. We prove the following. {\parindent=7mm \begin{itemize}\item[(1)]An \(OA(3, s)\) of \(\alpha\)-type exists if and only if \(s\) is a prime power. \item[(2)]\(OA(3, s)\)s of \(\alpha\)-type are isomorphic to each other as orthogonal arrays. \item[(3)]An \(OA(3, s)\) of \(\alpha\)-type yields \(PG(3, s)\). \item[(4)]The 3-dimensional Rao-Hamming is an \(OA(3, s)\) of \(\alpha\)-type. \item[(5)]A linear \(OA(3, s)\) is of \(\alpha\)-type. \end{itemize}}