  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.
