If $C$ is cyclic code with generator polynomial $g(x)$ and $D$ is subset of $C$ consisting of only even-weight vectors from $C$, what is generator polynomial of $D$ in terms of $g(x)$?
I can manage to show that $D$ is indeed cyclic code, but I don't know it's generator polynomial. I know it is polynomial of smallest degree that is in ideal which corresponds $D$, but how do I determine it in terms of $g(x)$?
I assume we are working over $\Bbb F_2$.
If all the code-words already have even weight then the generator polynomial remains $g(X)$.
Otherwise, consider $(1+X)g(X)$. That corresponds to an even-weight word, doesn't it? Can we have a polynomial of smaller degree than that giving rise to an even-weight word?