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

Abstract

Let $G$ be a finite abelian group. If $f: G\rightarrow {\mathbf{C}} $ is a nonzero function with Fourier transform $\hat {f}$ , the Donoho-Stark uncertainty principle states that $| {\mathrm{ supp}}(f)|| {\mathrm{ supp}}(\hat {f})|\geq |G|$ . 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\rightarrow {\mathbf{F}} $ is a non-zero function from $G$ to a field ${\mathbf{F}}$ , and if $f$ has a Fourier transform $\hat {f}$ , the sharpened uncertainty principle states that $| {\mathrm{ supp}}(f)|| {\mathrm{ supp}}(\hat {f})|\geq |G|+| {\mathrm{ supp}}(f)|-|H({\mathrm{ supp}}(f))|$ , where $H({\mathrm{ supp}}(f))$ is the stabilizer of ${\mathrm{ supp}}(f)$ in $G$ .