The weight distributions of cyclic codes with two zeros and zeta functions

We consider the weight distribution of the binary cyclic code of length 2n−1 with two zeros αa,αb. Our proof gives information in terms of the zeta function of an associated variety. We carry out an explicit determination of the weight distribution in two cases, for the cyclic codes with zeros α3,α5 and α,α11. These are the smallest cases of two infinite families where finding the weight distribution is an open problem. Finally, an interesting application of our methods is that we can prove that these two codes have the same weight distribution for all odd n.