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

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.

16 pages