Abstract
In this paper, the dual code of the binary cyclic code of length 2n − 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 = 2n/2 + 1, t 2 = 2n−1 − 2n/2−1 + 1 or 2n/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.
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Li, C., Zeng, X. & Hu, L. A class of binary cyclic codes with five weights. Sci. China Math. 53, 3279–3286 (2010). https://doi.org/10.1007/s11425-010-4062-z
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DOI: https://doi.org/10.1007/s11425-010-4062-z