Primitive cyclic Reed-Solomon code ($C_{k,b}=⟨{\prod }_{\ell =b}^{b+n-k-1}(x-{\omega }^{\ell })⟩$)

Let $n=q-1$ and let $\omega$ be a primitive root of unity in ${\mathbb{F}}_q$. The length-$n$ cyclic code with consecutive zeros of a cyclic code ${\omega }^b,\dots ,{\omega }^{b+n-k-1}$ has dimension $k$. It equals the generalized Reed-Solomon code of vectors $({\omega }^{j(1-b)}f({\omega }^j){)}_{j=0}^{n-1}$, $\mathrm{deg}f<k$. The choice $b=1$ is the unweighted Reed-Solomon code on these evaluation points; allowing other offsets makes this family closed under dual codes.