On binary Kloosterman sums divisible by 3

K. Garaschuk, P. Lisonek,
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_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 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.

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