Reciprocal-polynomial parity-check matrix
= 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 $h^*,Xh^*,\ldots,X^{r-1}h^*$ 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 $X^{N-r},\ldots,X^{N-1}$ in its product with $h$, which vanish for codewords.
// Target: algebra.bigb