Duality of primitive cyclic Reed-Solomon codes ($C_{k,b}^{\perp }=C_{n-k,1-b}$)

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.