Check polynomial of a cyclic code ($h(X)=(X^N-1)/g(X)$)

For a cyclic code with monic generator $g\mid X^N-1$, the canonical check polynomial is $h=(X^N-1)/g$. A word $a$ belongs to the code exactly when $ah$ is zero modulo $X^N-1$, since cancellation in the polynomial ring turns this condition into $g\mid a$. The kernel condition alone allows additional polynomials: it is equivalent to $\gcd (\tilde{h},X^N-1)=h$. Requiring the monic divisor of the modulus singles out the canonical choice.

// Target: algebra.bigb