= Primitive cyclic Reed-Solomon code
{title2=$C_{k,b}=\langle\prod_{\ell=b}^{b+n-k-1}(x-\omega^\ell)\rangle$}
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> $\omega^b,\ldots,\omega^{b+n-k-1}$ has dimension $k$. It equals the <generalized Reed-Solomon code> of vectors $(\omega^{j(1-b)}f(\omega^j))_{j=0}^{n-1}$, $\deg f<k$. The choice $b=1$ is the unweighted <Reed-Solomon code> on these evaluation points; allowing other offsets makes this family closed under <dual codes>.
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