# On the weight structure of cyclic codes over $GF(q)$, $q>2$

## TL;DR

This paper explores the weight structure of cyclic codes over GF(q) for q>2, establishing relations with their proportional elements, and determines conditions for equidistance and minimum distance in certain nonprimitive codes.

## Contribution

It introduces a new approach linking cyclic code structure with proportional elements and provides new results on equidistance and minimum distance for nonprimitive codes.

## Key findings

- Derived conditions for equidistance of irreducible nonprimitive codes
- Identified minimum distances for specific classes of nonprimitive cyclic codes
- Established the relation between cyclic structure and weight distribution

## Abstract

The interrelation between the cyclic structure of an ideal, i.e., a cyclic code over Galois field $GF(q)$, $q>2$, and its classes of proportional elements is considered. This relation is used in order to define the code's weight structure. The equidistance conditions of irreducible nonprimitive codes over GF(q) are given. Besides that, the minimum distance for some class of nonprimitive cyclic codes is found.