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{\bf Kazuaki Ishii}
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{\bf Orthogonal Arrays with Parameters $OA(s^3,s^2+s+1,s,2)$ and 3-Dimensional Projective Geometries}
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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
$OA(s^n, (s^n-1)/(s-1),s,2)$. 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.\\
(1) An $OA(3,s)$ of $\alpha$-type exists if and only if $s$ is a prime power.\\
(2) $OA(3,s)$s of $\alpha$-type are isomorphic to each other as orthogonal arrays.\\
(3) An $OA(3,s)$ of $\alpha$-type yields $PG(3,s)$.\\
(4) The 3-dimensional Rao-Hamming is an $OA(3,s)$ of $\alpha$-type.\\
(5) A linear $OA(3,s)$ is of $\alpha $-type. \\
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