A class of binary cyclic codes with five weights

In this paper, the dual code of the binary cyclic code of length 2 ⁿ ? 1 with three zeros α, $ \alpha ^{t_1 } $ and $ \alpha ^{t_2 } $ is proven to have five nonzero Hamming weights in the case that n ? 4 is even and t ₁ = 2 ^n/2 + 1, t ₂ = 2 ^n?1 ? 2 ^n/2?1 + 1 or 2 ^n/2 + 3, where α is a primitive element of the finite field $ \mathbb{F}_{2^n } $ . The dual code is a divisible code of level n/2 ?1, and its weight distribution is also completely determined. When n = 4, the dual code satisfies Ward’s bound.