On cyclic codes of length ${2^{2^r}-1}$ with two zeros whose dual codes have three weights

Abstract

In 1976, Helleseth conjectured that two binary m-sequences of length 2^m− 1 can not have a three-valued crosscorrelation function when m is a power of 2. We show that this conjecture is true when −1 is a correlation value. In other words, if ${{\mathcal{C}}_{1,k}}$ is the cyclic code of length 2^m− 1 with two zeros α, α ^k, where α is a primitive element of ${{\mathbb{F}}_{2^m}}$ and gcd(k, 2^m − 1) = 1, then its dual ${{\mathcal{C}}_{1,k}^{\perp}}$ can not have three weights when m is a power of 2.