Harmonic analysis and a bentness-like notion in certain finite Abelian groups over some finite fields

TL;DR

This paper develops a harmonic analysis framework for certain finite Abelian groups over finite fields, introduces a new bent function concept for finite field-valued functions, and explores their properties and generalizations.

Contribution

It introduces a novel bentness notion for finite field-valued functions and extends harmonic analysis tools to specific finite Abelian groups over finite fields.

Findings

Established a modular character theory for these groups

Defined and analyzed properties of the new bent functions

Connected the new bentness concept to existing theories

Abstract

It is well-known that degree two finite field extensions can be equipped with a Hermitian-like structure similar to the extension of the complex field over the reals. In this contribution, using this structure, we develop a modular character theory and the appropriate Fourier transform for some particular kind of finite Abelian groups. Moreover we introduce the notion of bent functions for finite field valued functions rather than usual complex-valued functions, and we study several of their properties. In particular we prove that this bentness notion is a consequence of that of Logachev, Salnikov and Yashchenko, introduced in "Bent functions on a finite Abelian group" (1997). In addition this new bentness notion is also generalized to a vectorial setting.