Divisibility properties of classical binary Kloosterman sums

Let K ( a ) be the so-called classical Kloosterman sum over F 2 m . In this paper, we compute K ( a ) modulo 24 for even m , completing our previous results for odd m . We extensively study the links between K ( a ) and other exponential sums, especially the cubic sums. We point out (as we did for odd m ) that the values K ( a ) are involved in the computation of the weight distributions of cosets of primitive narrow sense extended BCH codes of length 2 m and minimum distance 8. We also complete some recent results on K ( a ) − 1 modulo 3.