Solution
= Solution
A <cyclic code>[binary cyclic code] is a linear subspace $C\leq\mathbb F_2^n$ invariant under cyclic coordinate shift. Identify a word $(c_0,\ldots,c_{n-1})$ with $c_0+c_1X+\cdots+c_{n-1}X^{n-1}$ in
$$
\mathbb F_2[X]/(X^n-1).
$$
Cyclic shift is multiplication by $X$, so cyclic codes are exactly ideals of this quotient. Since $\mathbb F_2[X]$ is a <principal ideal domain>, each is generated by a unique monic divisor $g\mid X^n-1$, giving the required bijection.