This paper considers the weight distributions of binary cyclic codes with one or two zeros. Results on value distributions of certain monomial exponential sums and connections between exponential sums and Kloosterman sums are applied together with the Pless power moment identity. Explicit formulae for the number of low-weight codewords are given as well as the entire weight distribution for some classes of codes. We show that the minimum distance of the cyclic codes with one zero, i.e., the duals of irreducible cyclic codes, in so-called index 2 case is three except for two codes.

irreducible cyclic codes, weight distribution, Gauss sums, Kloosterman sums, Pless power moment identity.