In the discuss with @Evinda we have the contradiction with the definition of Reed-Solomon Codes (not generalized case) over the finite field $\mathbb{F}_q$. We have two papers
and
http://kom.aau.dk/~heb/kurser/NOTER/KOFA01.PDF
The first say that the length of $RS$ code is $q-1$ and the second say that is equal to $q$. Where is the correct definition? Thanks
Generally speaking, the length $n$ of RS code $\mathcal{K}$ over $\mathbb{F}_q$ is not $q-1$. In the case, when $n=q-1$ we obtain cyclic RS-code. We can define RS-code also as a linear recursive code over $\mathbb{F}_q$. Let $\omega$ be a primitive element of $\mathbb{F}_q$ and $f(x) = (x-\omega^s)(x-\omega^{s+1})\ldots(x-\omega^{s+n-1})$, where $2\leq s\leq q-1$ and $2\leq n\leq q-1$, then $\mathcal{K} = L_{\mathbb{F}_q}^{\overline{0,n-1}}(f(x))$. That is, $\mathcal{K}$ is the set of all an $n$-segments of linear recurrences over $\mathbb{F}_q$ with characteristic polynomial $f(x)$.