# Non-projective cyclic codes whose check polynomial contains two zeros

## TL;DR

This paper proves that a specific family of non-projective cyclic codes with two zeros in their check polynomial cannot be two-weight codes, supporting Vega's conjecture that all two-weight cyclic codes are already known.

## Contribution

It establishes that these particular cyclic codes are never two-weight, providing evidence for the conjecture that all two-weight cyclic codes are classified.

## Key findings

- These codes are never two-weight codes.
- Supports Vega's conjecture on the classification of two-weight cyclic codes.

## Abstract

Let $n\geq 3$ 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.