On a Grouping Method for Constructing Mixed Orthogonal Arrays

Mixed orthogonal arrays of strength two and size s are constructed by grouping points in the finite projective geometry . can be partitioned into   1, PG mn s   1, PG mn s       1 1 mn n s s         –1 n -flats such that each   –1 n flat is associated with a point in . An orthogonal array  1, n PG m s        1 1 mn n mn s s n s L s         can be constructed by using     1 1 s  mn n s  points in  1, n  PG m s  . A set of     1 1 s t s   points in   n 1, PG m  s is called a   –1 t flat over GF(s) if it is isomorphic to . If there exists a  1, PG t s     –1 t -flat over GF(s) in , then we can replace the corresponding  1, n m s   PG     1 1  t s s     s-level columns in       1 1 mn n mn s s n s L s         by     1 1 s s       n s-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   –1 t -flats over GF(s) in 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  1, PG m s   n         1 1 1 i s m s  1 i s s  1 n m