The uncertainty principle over finite fields

07/08/2020

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by   Martino Borello, et al.

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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. The naive version is the direct analogue over finite fields of the Donoho-Stark bound over the complex numbers. A connection with Ramsey Theory is pointed out. It is strong enough to show that there exist sequences of cyclic codes of length n, arbitrary rate, and minimum distance Ω(n^α) for all 0<α<1/2. The strong version for a given finite field 𝔽_q is shown to hold only for primes p>q+2 for which q is primitive, depending on the MDS Conjecture. The weak version is shown to be impossible in some cases.