Title: Good cyclic codes and the uncertainty principle,
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.