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

Abstract

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 $m\ge 5$ 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.

29 pages