R-orthogonality of Latin Squares using Permutation Polynomials

A latin square of order n is a n x n matrix, where the entries are n distinct elements and the elements in each row and each column do not repeat. Latin squares have applications in coding theory, projective geometry, and other fields of discrete mathematics. In our research we study the r-orthogonality of latin squares whose first column is given by permutation polynomials.  We conjecture that the r-orthogonality of two latin squares generated by permutation polynomials can be computed using only the permutation used to construct the latin squares. Let LS₁ and LS₂ be latin squares or order p generated using permutation polynomials. Define: D = {(P₁(l)- P₂(l)) mod p | 0 ≤ l ≤ (n-1)} and k = |D|, then LS₁ and LS₂ are (k)(p)-orthogonal. This conjecture saves a great amount of computational time. Currently we are computing examples of our conjecture using an algorithm implemented in C++, and the examples we have computed confirm our conjecture. However, as this is a work in progress, we are also working towards a mathematical proof of our conjecture using the properties of permutation polynomials.