Let $C_1, C_2$ be two cyclic codes over $GF(q)$ and $g_1(x), g_2(x)$ their generator polynomials. $C_1$ is of [n,k], $C_2$ is of [n,m].
What is the generator polynomial of $C_1 \cap C_2$? I'm think the answer should be somewhere around $lcm(g_1, g_2)$ But so far I failed to show that it generates the codewords that exist both in $C_1$ and $C_2$.