  In this paper we study the uncertainty principle (UP) connecting a function
over a finite field and its Mattson-Solomon polynomial, which is a kind of
Fourier transform in positive characteristic. Three versions of the UP over
finite fields are studied, in connection with the asymptotic theory of cyclic
codes. We first show that no finite field satisfies the strong version of UP,
introduced recently by Evra, Kowalsky, Lubotzky, 2017. A refinement of the weak
version is given, by using the asymptotic Plotkin bound. A naive version, which
is the direct analogue over finite fields of the Donoho-Stark bound over the
complex numbers, is proved by using the BCH bound. It is strong enough to show
that there exist sequences of cyclic codes of length $n$, arbitrary rate, and
minimum distance $\Omega(n^\alpha)$ for all $0<\alpha<1/2$. Finally, a
connection with Ramsey Theory is pointed out.
