  Based on a sufficient condition proposed by Hollmann and Xiang for
constructing triple-error-correcting codes, the minimum distance of a binary
cyclic 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 code
are studied, where $m\geq 5$ is odd and $\alpha$ is a primitive element of the
finite field $\mathbb{F}_{2^m}$. The code $\mathcal{C}_{1,3,13}$ is proven to
have the same weight distribution as the binary triple-error-correcting
primitive BCH code $\mathcal{C}_{1,3,5}$ of the same length.
