The GC-content of a family of cyclic codes with applications to DNA-codes
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Given a prime power q and a positive integer r>1 we say that a cyclic code of length n, C⊆ F_q^r^n, is Galois supplemented if for any non-trivial element σ in the Galois group of the extension F_q^r/ F_q, C+C^σ= F_q^r^n, where C^σ={(x_1^σ,...,x_n^σ)| (x_1,...,x_n)∈ 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.
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