Binary Cyclic Codes with Two Primitive Nonzeros

In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of Tr(x d ) over \(\mathbb{F}_{2^m }\) in the case where m = 2t, d = 3+2 t+1 and gcd(d, 2 m − 1) = 1.