On a Grouping Method for Constructing Mixed Orthogonal Arrays
- 1 Department of Mathematics, Cleveland State University, Cleveland, USA
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)/( 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 s mn (( s n ) ( s mn -)( s n -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 s mn (( s n ) ( s mn -)( s n -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 s mn (( s m ) s mn -1/ s m -1-i( s n -1)/ ( s -1) ( s n ) i( s m -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.
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