The uncertainty principle over finite fields

TL;DR

This paper explores the uncertainty principle over finite fields, analyzing different versions and their implications for cyclic codes, and establishes bounds and connections to Ramsey Theory.

Contribution

It introduces and compares three versions of the uncertainty principle over finite fields, providing bounds and linking to coding theory and Ramsey Theory.

Findings

No finite field satisfies the strong UP version.

A refined weak UP version is established using the Plotkin bound.

Existence of cyclic codes with specific rate and minimum distance is demonstrated.

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.