Rank-deficient submatrices of Fourier matrices

AbstractWe consider the maximal rank-deficient submatrices of Fourier matrices with order a power of a prime number. We do this by considering a hierarchical subdivision of these matrices into low rank blocks. We also explore some connections with the fast Fourier transform (FFT), and with an uncertainty principle for Fourier transforms over finite Abelian groups

A new inequality in discrete Fourier theory

Discrete Fourier theory has been applied successfully in digital communication theory. In this correspondence, we prove a new inequality linking the number of nonzero components of a complex valued function defined on a finite Abelian group to the number of nonzero components of its Fourier transform. We characterize the functions achieving equality. Finally, we compare this inequality applied to Boolean functions to the inequality arising from the minimal distance property of Reed-Muller codes.status: publishe