= Root-evaluation parity-check matrix of a cyclic code
For a <cyclic code> whose <generator polynomial> has distinct roots $\alpha_1,\ldots,\alpha_r$ in a <splitting field> $E$, the rows $(1,\alpha_i,\ldots,\alpha_i^{n-1})$ 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.
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