Secondary constructions of vectorial $p$-ary weakly regular bent functions

TL;DR

This paper generalizes methods for constructing vectorial p-ary weakly regular bent functions, introduces new infinite families, and offers a new characterization of the $(P_U)$ property using second-order derivatives.

Contribution

It extends existing secondary construction techniques for p-ary bent functions and provides a novel characterization of the $(P_U)$ property in the p-ary setting.

Findings

Constructed new infinite families of vectorial p-ary weakly regular bent functions.

Generalized the $(P_U)$ property characterization via second-order derivatives.

Extended secondary construction methods to the p-ary case.

Abstract

In \cite{Bapic, Tang, Zheng} a new method for the secondary construction of vectorial/Boolean bent functions via the so-called $(P_U)$ property was introduced. In 2018, Qi et al. generalized the methods in \cite{Tang} for the construction of $p$-ary weakly regular bent functions. The objective of this paper is to further generalize these constructions, following the ideas in \cite{Bapic, Zheng}, for secondary constructions of vectorial $p$-ary weakly regular bent and plateaued functions. We also present some infinite families of such functions via the $p$-ary Maiorana-McFarland class. Additionally, we give another characterization of the $(P_U)$ property for the $p$-ary case via second-order derivatives, as it was done for the Boolean case in \cite{Zheng}.