Conference signals: Applications, existence, and constructions

Source abstract

A conference signal is a complex-valued function on a finite abelian group that vanishes at 0 in both the time and frequency domains, and is otherwise flat in both domains. For example, the Legendre symbol is a conference signal on $\mathbb{Z}_p$ since it vanishes at $0$, takes values $\pm 1$ away from zero, and is a scalar multiple of its Fourier transform. More generally, any nontrivial multiplicative character on a finite field is a conference signal on its additive group. Until now, conference signals have mainly been studied by number theorists, who wanted to know if multiplicative characters were the only conference signals on finite fields. The state of the art uses fancy tools from symplectic geometry to prove that non-character conference signals exist on non-small fields. The same tools show conference signals exist on every abelian group of odd order, without saying what they are. In this (human-friendly!) paper, we explain how conference signals produce other objects of interest in algebraic combinatorics and applied harmonic analysis, and then we present some new, explicit constructions of conference signals on $\mathbb{Z}_n$. In particular, we give the first explicit conference signals for composite $n$ and the first explicit non-character examples for prime $n$.