Repeated-root parity-check matrix

If the generator polynomial of a cyclic code has distinct roots ${\alpha }_i$ with multiplicities $m_i$, a word belongs to the code exactly when $c^{[\ell ]}({\alpha }_i)=0$ for $0\le \ell <m_i$, where $c^{[\ell ]}$ is its Hasse derivative. The corresponding rows have entries $\left(\genfrac{}{}{0ex}{}{j}{\ell }\right){\alpha }_i^{j-\ell }$ for $j\ge \ell$ and zero otherwise. Expansion in a basis of the splitting field gives equations over the original finite field; an independent selection has rank ${\sum }_i{m}_i$.