Suppose that we have a binary cyclic code $C$ of length $n \geq 3$ with generator polynomial $b(x) \neq 1$, where $n$ is the smallest natural number such that $b(X) \mid X^n-1$. I want to show that the minimum distance of the code is at least $3$.
How can we get information about the distance of the code, although we do not have a specific $n$, i.e. we don't have the factorization into irreducibles of $X^n-1$ ?
If $c(X)$ is any codeword, then $b(X)\mid c(X)$. But if $c(X)$ has weight two, then $c(X)=X^a+X^b$ for some integers $a,b$, $0\le a<b<n$. So $c(X)=X^a(X^{b-a}-1)$. This means that $$ \gcd(c(X),X^n-1)=\gcd(X^{b-a}-1,X^n-1)=X^{\gcd(b-a,n)}-1. $$ Combine this with the fact that $b(X)\mid \gcd(c(X),X^n-1)$.