Mutually orthogonal Latin squares via polynomials modulo \(n\) (Q2839651)

Let \(L\) be an \(n \times n\) Latin square whose row and column indices, \(x\) and \(y\), are elements of \(Z_n\). Define \(L\) to be polynomial generated if there is a polynomial function, \(f(x,y)\) of \(x\) and y such that \(L(x,y) = f(x,y)\) for all \(x\) and \(y\). The authors show that for any integer \(n \geq 2\), the maximum number of polynomial generated, mutually orthogonal Latin squares of order \(n\) is \(p-1\), where \(p\) is the smallest prime dividing \(n\).