Generator polynomial of the sum of cyclic codes

Question

Given two cyclic codes $C_1$ and $C_2$ of length $n$ with generator polynomials $g_1(X)$ and $g_2(X)$ respectively, I have to find the generator polynomial of $C_1+C_2$, which I already know that is a cyclic code.

I've thought of $g(X)=\gcd(g_1(X),g_2(X))$ as an answer due to the fact that $C_i\subset C_1+C_2$, and therefore the answer should generate both codes. It's obvious that any element of $C_1+C_2$ is spanned by linear combinations of $X^ig(X)$, but I need to show that this does not generate anything else.

My idea is to build the basis $\{g(X), Xg(X),\dots, X^{n-r-1}g(X)\}$ where $r=\deg(g(X))$ and show that it is indeed a basis. Would it be the right way?

Answer 1

Every codeword in $C_1+C_2$ can be expressed as $a_1(x)g_1(x) + a_2(x)g_2(x)$. Now think of Bezout's theorem. What is the polynomial of least degree that can be expressed in this form?

Answer 2

Since enough time has passed since the original posting of this question that this response doesn't spoil the learning process for anyone, I will add the following result for anyone who stumbles across this looking for more information about $C_1 + C_2$. (Unfortunately, I don't have enough reputation points to add this as a comment.)

Let consider a let $n$ be a positive integer and $p$ a prime (WLOG) and assume $gcd(p, n) = 1$. The generator matrix of a cyclic code is a factor $x^n - 1 = g(x) h(x)$ over the splitting field $GF(p^m)$ where $m = ord_n(q)$. Let $\alpha$ be a primitive $n$-th root of this field such that the roots of $g(x)$ are ${\alpha^{t_1}, ..., \alpha^{t_r}}$. The defining set of the code is the set $T = {t_1, \dots, t_r}$. In addition to $g(x)$, the code may also be generated by a polynomial $e(x)$ called the generator idempotent. This is determined by using the extended Euclidean algorithm to find $a(x)$ and $b(x)$ such that $1 = a(x) g(x) + b(x) h(x)$; then $e(x) = a(x) g(x)$.

Extending the posted question, given two cyclic codes $C_1$ and $C_2$, $C_1 + C_2$ has generator polynomial $\gcd(g_1(x), g_2(x))$, generator idempotent $e_1(x) + e_2(x) - e_1(x) e_2(x)$, and defining set $T_1 \cap T_2$. More information may be found in standard texts covering cyclic codes (e.g. Huffman and Pless is an understandable one with modern mathematical notation).