The Shift Bound for Abelian Codes and Generalizations of the Donoho-Stark Uncertainty Principle

Let G be a finite abelian group. If f : G -> C is a nonzero function with Fourier transform f, the Donoho-Stark uncertainty principle states that |supp( f)||supp((f) over cap)| >= |G|. The purpose of this paper is twofold. First, we present the shift bound for abelian codes with a streamlined proof. Second, we use the shifting technique to prove a generalization and a sharpening of the Donoho-Stark uncertainty principle. In particular, if f : G -> F is a non-zero function from G to a field F, and if f has a Fourier transform (f) over cap, the sharpened uncertainty principle states that |supp(f)|| supp((f) over cap)| >= | G|+| supp(f)|-| H(supp(f))|, where H(supp(f)) is the stabilizer of supp(f) in G.