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On a Grouping Method for Constructing Mixed Orthogonal Arrays

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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)/(*s*^{n}-1)](n-1)-flats such that each (n-1)-flat is associated with a point in PG(m-1,*s*^{n}). An orthogonal array*L*_{smn}((*s*^{n})^{(smn-)(sn-1)}can be constructed by using (*s*^{mn}-1)/(*s*^{n}-1) points in PG(m-1,*s*^{n}). A set of (st-1)/(s-1) points in PG(m-1,*s*^{n}) 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,*s*^{n}), then we can replace the corresponding [(st-1)/(s-1)] sn-level columns in*L*_{smn}((*s*^{n})^{(smn-)(sn-1)}by (*s*^{mn}-1)/(*s*^{n}-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,*s*^{n}) 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*_{smn}((*s*^{m})^{smn-1/sm-1-i(sn-1)/ (s-1)}(*s*^{n})^{ i(sm-1)/ s-1}) for any 0__<__*i*__<__(*s*^{mn}-1)(*s*-1)/(*s*^{m}-1)(*s*^{n}-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

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The authors declare no conflicts of interest.

Cite this paper

C. Suen, "On a Grouping Method for Constructing Mixed Orthogonal Arrays,"

*Open Journal of Statistics*, Vol. 2 No. 2, 2012, pp. 188-197. doi: 10.4236/ojs.2012.22022.

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