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Let $n\geq 3$ be a positive integer and let $\mathbb{F}_{q^k}$ be thesplitting 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 checkpolynomial 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 certainparameters. In this paper, we prove that these codes are never two-weightcodes. This result would strengthen a conjecture by Vega which states that alltwo-weight cyclic codes are the "known" ones.

Non-projective cyclic codes whose check polynomial contains two zeros