The uncertainty principle over finite fields

Abstract: 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.