On binary and ternary Kloosterman sums

We study exponential sums K(a), a is in GF(2^m) or GF(3^m), known as Kloosterman sums. For the binary case we establish the exact spectrum of the number of coset leaders of weight 3 of the binary Melas code. We derive a family of elliptic curves that allows us to characterize all a in GF(2^m) for which K(a) is divisible by 3. As an application we construct so-called "caps with many free pairs of points" in PG(n,2) and describe their use in statistical experimental designs. In the ternary case, by transforming a certain system of equations over GF(3^m)\{0} into a parametrized family of elliptic curves, we classify and count those a in GF(3^m) for which K(a)=0,2 (mod 4). We also present a result which is of independent interest, namely a generalization of the well known fact that tr(a)=0 (a in GF(2^m)) if and only if a=t^2+t for some a in GF(2^m).