# The GC-content of a family of cyclic codes with applications to DNA-codes

## TL;DR

This paper introduces Galois supplemented cyclic codes over finite fields, generalizing quadratic-residue codes, and explores their properties and applications to DNA coding, especially regarding GC-content control.

## Contribution

The paper defines Galois supplemented codes, extends properties of QR-codes to this family, and provides explicit counting methods for words with fixed coordinates, with applications to DNA coding.

## Key findings

- Galois supplemented codes include QR-codes over F_{q^2}.
- Explicit enumeration of codewords with fixed F_q-coordinates is achieved.
- Applications to DNA codes with controlled GC-content are demonstrated.

## Abstract

Given a prime power $q$ and a positive integer $r>1$ we say that a cyclic code of length $n$, $C\subseteq F_{q^r}^n$, is Galois supplemented if for any non-trivial element $\sigma$ in the Galois group of the extension $ F_{q^r}/ F_q$, $C+C^\sigma= F_{q^r}^n$, where $C^\sigma=\{(x_1^\sigma,\dots,x_n^\sigma)\mid (x_1,\dots,x_n)\in C\}$. This family includes the quadratic-residue (QR) codes over $ F_{q^2}$. Some important properties QR-codes are then extended to Galois supplemented codes and a new one is also considered, which is actually the motivation for the introduction of this family of codes: in a Galois supplemented code we can explicitly count the number of words that have a fixed number of coordinates in $ F_q$. In connection with DNA-codes the number of coordinates of a word in $ F_4^n$ that lie in $ F_2$ is sometimes referred to as the $GC$-content of the word and codes over $ F_4$ all of whose words have the same $GC$-content have a particular interest. Therefore our results have some direct applications in this direction.