Unimodular Matrices and Parsons Numbers

Let [A1 , ..., Am] be a set of m matrices of size n_n over the field F such that Ai # SL(n, F) for 1 i m and such that Ai&Aj # SL(n, F) for 1 i< j m. The largest integer m for which such a set exists is called the Parsons number for n and F, denoted m(n, F). We will call such a set of m(n, F) matrices a Parsons set: such a set arises in a combinatorial setting (see [Z]). Parsons asserted (see [Z]) that m(n, Fq) q if Fq is the Galois field of order q. Here we will consider the case n=2. Our result is the following.