Weil-Serre type bounds for cyclic codes

We give a new method in order to obtain Weil-Serre type bounds on the minimum distance of arbitrary cyclic codes over $\mathbb{F}_{p^e}$ of length coprime to $p$, where $e\geq 1$ is an arbitrary integer. In an earlier paper we obtained Weil-Serre type bounds for such codes only when $e = 1$ or $e = 2$ using lengthy explicit factorizations, which seems hopeless to generalize. The new method avoids such explicit factorizations and it produces an effective alternative. Using our method we obtain Weil-Serre type bounds in various cases. By examples we show that our bounds perform very well against Bose-Chaudhuri-Hocquenghem bound and they yield the exact minimum distance in some cases.