TAILIEUCHUNG - Báo cáo toán học: "Orthogonal arrays with parameters OA(s3, s2+s+1, s, 2) and 3-dimensional projective geometries"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Orthogonal arrays with parameters OA(s3, s2+s+1, s, 2) and 3-dimensional projective geometries. | Orthogonal arrays with parameters OA s3 s2 s 1 s 2 and 3-dimensional projective geometries Kazuaki Ishii Submitted Feb 26 2010 Accepted Mar 22 2011 Published Mar 31 2011 Mathematics Subject Classification 05B15 Abstract 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 sn sn 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 a-type. We prove the following. 1 An OA 3 s of a-type exists if and only if s is a prime power. 2 OA 3 s s of a-type are isomorphic to each other as orthogonal arrays. 3 An OA 3 s of a-type yields PG 3 s . 4 The 3-dimensional Rao-Hamming is an OA 3 s of a-type. 5 A linear OA 3 s is of a-type. Keywords orthogonal array projective space projective geometry 1 Introduction An N X k array A with entries from a set S that contains s symbols is said to be an orthogonal array with s levels strength t and index A if every N X t subarray of A contains Osaka prefectual Nagano high school 1-1-2 Hara Kawachinagano Osaka Japan e-mail denen482@ THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P69 1 each t tuple based on S exactly A times as a row. We denote the array A by OA N k s t . Orthogonal arrays with parameters OA sn sn 1 s 1 s 2 are known for any prime power s and any integer n 2. For example orthogonal arrays of Rao-Hamming type have such parameters. We are interested in whether orthogonal arrays with .

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