Several classes of bent, near-bent and 2-plateaued functions over finite fields of odd characteristic

TL;DR

This paper introduces new infinite families of quadratic ternary bent, near-bent, and 2-plateaued functions over finite fields of odd characteristic, expanding the known classes and analyzing their Walsh spectra.

Contribution

It presents novel infinite families of quadratic ternary bent, near-bent, and 2-plateaued functions, including the first $p$-ary bent functions of algebraic degree $p$ over odd characteristic fields.

Findings

Distribution of Walsh spectrum for two classes of 2-plateaued functions is fully determined.

Constructs the first class of $p$-ary bent functions of algebraic degree $p$ over arbitrary odd characteristic fields.

Includes non-quadratic $p$-ary bent functions that are affinely inequivalent to known functions.

Abstract

Inspired by a recent work of Mesnager, we present several new infinite families of quadratic ternary bent, near-bent and 2-plateaued functions from some known quadratic ternary bent functions. Meanwhile, the distribution of the Walsh spectrum of two class of 2-plateaued functions obtained in this paper is completely determined. Additionally, we construct the first class of $p$-ary bent functions of algebraic degree $p$ over the fields of an arbitrary odd characteristic. The proposed class contains non-quadratic $p$-ary bent functions that are affinely inequivalent to known monomial and binomial ones.