Divisibility properties of Kloosterman sums over finite fields of characteristic two

Let K(a) be the so-called classical Kloosterman sums over F₂m, where m is even. In this paper, we compute K(a) modulo 24, completing our previous results for odd m. We extensively study the links between K(a) and other exponential sums, in particular with the cubic sums. We point out (as we did for odd m) that the values K(a) are related with cosets of weight 4 of primitive narrow sense extended BCH codes of length n = 2^m and minimum distance 8.