Non-projective cyclic codes whose check polynomial contains two zeros

Abstract

Let $n\ge 3$ be a positive integer and let ${\mathbb{F}}_{q^k}$ be the splitting field of $x^n-1$. By $\gamma$ we denote a primitive element of ${\mathbb{F}}_{q^k}$. Let $C$ be a cyclic code of length $n$ whose check polynomial contains two zeros ${\gamma }^d$ and ${\gamma }^{d+D}$, where $de\mid (q-1)$, $e>1$ and $D=(q^k-1)/e$. This family of cyclic codes is not projective. Many authors have studied the weight distribution of these codes for certain parameters. In this paper, we prove that these codes are never two-weight codes. This result would strengthen a conjecture by Vega which states that all two-weight cyclic codes are the "known" ones.