On binary Kloosterman sums divisible by 3.

Designs, Codes and Cryptography 49 (2008), 347-357.

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 F_{2m} where m is odd.
New results due to Charpin, Helleseth
and Zinoviev then provide a connection
to a characterization
of all a in F_{2m} such that
Tr(a^{1/3})=0; we prove
a generalization to the case Tr(a^{1/(2k-1)})=0.
We present an application to constructing caps in PG(n,2)
with many free pairs of points.