This paper considers rational integer values of Kloosterman sums over finite fields of characteristicp> 3. We shall prove two main results. The first one is a congruence relation satisfied by possible integer values. One consequence is that there are no Kloosterman zeroes in the case of characteristicp> 3, which generalizes recent works by Shparlinski, Moisio, and Lisoněk on this subject. This, in turn, implies that there are no Dillon type bent functions in the casep> 3 , thus answering a question posed recently by Helleseth and Kholosha. Our other main result states that the Kloosterman sum obtains an integer value at a point if and only if the same sum lifted to any extension field remains an integer.

Two recursive algorithms for computing the weight distributions of certain binary irreducible cyclic codes of length n in the so-called index 2 case are presented. The running times of these algorithms are smaller than O(log/sup 2/r) where r=2/sup m/ and n is a factor of r-1.