Doubly Orthogonal Equi-Squares and Sliced Orthogonal Arrays

We introduce doubly orthogonal equi-squares. Using linear algebra over finite fields, we produce large families of mutually $q^s$-doubly orthogonal equi-$q^{r+s}$ squares, and show these are of maximal size when $s\le r+1$. These results specialize to the results of Xu, Haaland, and Qian when $r=s=1$ and the equi-squares are Sudoku Latin squares of order $q^2$. Further, we show how a collection of mutually $q^s$-doubly orthogonal equi-$q^{r+s}$ squares can be used to construct sliced orthogonal arrays of strength two. These orthogonal arrays have important applications in statistical designs.