Abstract
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.
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Garaschuk, K., Lisoněk, P. On binary Kloosterman sums divisible by 3. Des. Codes Cryptogr. 49, 347–357 (2008). https://doi.org/10.1007/s10623-008-9171-0
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DOI: https://doi.org/10.1007/s10623-008-9171-0