On a Grouping Method for Constructing Mixed Orthogonal Arrays

Chung-Yi Suen

Department of Mathematics, Cleveland State University, Cleveland, USA.

DOI: 10.4236/ojs.2012.22022                

Abstract

Mixed orthogonal arrays of strength two and size s^mn are constructed by grouping points in the finite projective geometry PG(mn-1, s). PG(mn-1, s) can be partitioned into [(s^mn-1)/(sⁿ-1)](n-1)-flats such that each (n-1)-flat is associated with a point in PG(m-1, sⁿ). An orthogonal array L_s^mn((sⁿ)^(smn-)(sn-1) can be constructed by using (s^mn-1)/( sⁿ-1) points in PG(m-1, sⁿ). A set of (st-1)/(s-1) points in PG(m-1, sⁿ) is called a (t-1)-flat over GF(s) if it is isomorphic to PG(t-1, s). If there exists a (t-1)-flat over GF(s) in PG(m-1, sⁿ), then we can replace the corresponding [(st-1)/(s-1)] sn-level columns in L_s^mn((sⁿ)^(smn-)(sn-1) by (s^mn-1)/( sⁿ-1) s^t -level columns and obtain a mixed orthogonal array. Many new mixed orthogonal arrays can be obtained by this procedure. In this paper, we study methods for finding disjoint (t-1)-flats over GF(s) in PG(m-1, sⁿ) in order to construct more mixed orthogonal arrays of strength two. In particular, if m and n are relatively prime then we can construct an L_s^mn((s^m)^{smn-1/sm-1-i(sn-1)/ (s-1)}( sⁿ) ^{i(sm-1)/ s-1}) for any 0<i<(s^mn-1)(s-1)/( s^m-1)( sⁿ-1) New orthogonal arrays of sizes 256, 512, and 1024 are obtained by using PG(7,2), PG(8,2), and PG(9,2) respectively.

Keywords

Finite field; Finite projective geometry; (t-1)-flat over GF(s) in PG(m-1, sⁿ ); Geometric orthogonal array; Matrix representation; Minimal polynomial; Orthogonal main-effect plan; Primitive element; Tight.

Conflicts of Interest

The authors declare no conflicts of interest.

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