Doubly Orthogonal Equi‐Squares and Sliced Orthogonal Arrays

ABSTRACT We introduce doubly orthogonal equi‐squares. Using linear algebra over finite fields, we produce large families of mutuallyq s‐doubly orthogonal equi‐q r + ssquares, and show these are of maximal size whens ≤ r + 1. These results specialize to the results of Xu, Haaland, and Qian whenr = s = 1and the equi‐squares are Sudoku Latin squares of orderq 2. Further, we show how a collection of mutuallyq s‐doubly orthogonal equi‐q r + ssquares can be used to construct sliced orthogonal arrays of strength two. These orthogonal arrays have important applications in statistical designs.