= Duality of primitive cyclic Reed-Solomon codes
{title2=$C_{k,b}^\perp=C_{n-k,1-b}$}
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 $\deg f<k$ and $\deg h<n-k$. 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.
Back to article page
