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Based on a sufficient condition proposed by Hollmann and Xiang forconstructing triple-error-correcting codes, the minimum distance of a binarycyclic code $\mathcal{C}_{1,3,13}$ with three zeros $\alpha$, $\alpha^3$, and$\alpha^{13}$ of length $2^m-1$ and the weight divisibility of its dual codeare studied, where $m\geq 5$ is odd and $\alpha$ is a primitive element of thefinite field $\mathbb{F}_{2^m}$. The code $\mathcal{C}_{1,3,13}$ is proven tohave the same weight distribution as the binary triple-error-correctingprimitive BCH code $\mathcal{C}_{1,3,5}$ of the same length.

A Triple-Error-Correcting Cyclic Code from the Gold and Kasami-Welch APN Power Functions