Root-evaluation parity-check matrix of a cyclic code

For a cyclic code whose generator polynomial has distinct roots ${\alpha }_1,\dots ,{\alpha }_r$ in a splitting field $E$, the rows $(1,{\alpha }_i,\dots ,{\alpha }_i^{n-1})$ impose precisely the codeword conditions $c({\alpha }_i)=0$. Expanding each value in a basis of $E$ over the original finite field, then discarding dependent equations, gives a parity-check matrix over that field of rank $r$. Alternatively choose one root from each Frobenius conjugate orbit and expand its value in the field it generates.