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.

With syndromes $S_{\ell }={\sum }_i{E}_i{X}_i^{b+\ell }$ and distinct error-location values $X_i$, form $S(z)={\sum }_{\ell =0}^{n-k-1}{S}_{\ell }{z}^{\ell }$ and the error locator polynomial $\Lambda (z)={\prod }_i(1-X_{i}z)$. There is an error evaluator polynomial $\Omega$ of degree less than the number of errors satisfying the key equation. When the number of errors is at most $\lfloor (n-k)/2\rfloor$, the Berlekamp-Massey algorithm or extended Euclidean algorithm recovers these polynomials, normalized by $\Lambda (0)=1$.

For $n=q-1$ and $a_j={\omega }^j$, the two evaluation forms are $a_j^{1-b}f(a_j)$ and $a_j^{b}h(a_j)$, with $\mathrm{deg}f<k$ and $\mathrm{deg}h<n-k$. Their dot product is ${\sum }_j{a}_{j}f(a_j)h(a_j)=0$, since every appearing exponent is between one and $n-1$ and the sums of those powers of $a_j$ vanish. Comparing dimensions proves equality with the dual code. In particular, the dual of the unweighted evaluation code has multipliers $a_j$; it need not be an unweighted evaluation code on the same ordered points.