# Good cyclic codes and the uncertainty principle

## TL;DR

This paper explores the connection between the uncertainty principle in harmonic analysis and the existence of good cyclic error-correcting codes, suggesting that a weak uncertainty principle could imply their existence.

## Contribution

It establishes a novel link between the uncertainty principle and cyclic codes, proposing that a weak form of the principle may guarantee the existence of good cyclic codes.

## Key findings

- A weak uncertainty principle implies the existence of good cyclic codes.
- Heuristic arguments support the potential validity of this implication.
- The work connects harmonic analysis principles with coding theory challenges.

## Abstract

A long standing problem in the area of error correcting codes asks whether there exist good cyclic codes. Most of the known results point in the direction of a negative answer. The uncertainty principle is a classical result of harmonic analysis asserting that given a non-zero function $f$ on some abelian group, either $f$ or its Fourier transform $\hat{f}$ has large support. In this note, we observe a connection between these two subjects. We point out that even a weak version of the uncertainty principle for fields of positive characteristic would imply that good cyclic codes do exist. We also provide some heuristic arguments supporting that this is indeed the case.