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

TL;DR

This paper constructs a new triple-error-correcting cyclic code using Gold and Kasami-Welch APN power functions, demonstrating its equivalence in weight distribution to a known BCH code.

Contribution

It introduces a novel triple-error-correcting cyclic code based on specific APN power functions and analyzes its minimum distance and weight distribution.

Findings

The code has the same weight distribution as the classical BCH code.

The minimum distance of the code is proven to be three.

The dual code's weight divisibility is characterized.

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 $m\geq 5$ 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.