On binary Kloosterman sums divisible by 3

By counting the coset leaders for cosets of weight 3 of the Melas code we give a new proof for the characterization of Kloosterman sums divisible by 3 for ${\mathbb{F}_{2^m}}$ where m is odd. New results due to Charpin, Helleseth and Zinoviev then provide a connection to a characterization of all ${a\in\mathbb{F}_{2^m}}$ such that ${Tr(a^{1/3})=0}$ ; we prove a generalization to the case ${Tr(a^{1/(2^k-1)})=0}$ . We present an application to constructing caps in PG(n, 2) with many free pairs of points.