Reciprocal-polynomial parity-check matrix

If a binary length-$N$ cyclic code has generator of degree $r$ and check polynomial $h$ of degree $N-r$, use the coefficient vectors of $h^{\ast },X{h}^{\ast },\dots ,X^{r-1}{h}^{\ast }$ as the rows of a parity-check matrix. They have distinct leading positions and are independent. Their scalar products with a word are the coefficients of $X^{N-r},\dots ,X^{N-1}$ in its product with $h$, which vanish for codewords.

// Target: algebra.bigb