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Some New Constructions of Generalized Plateaued Functions

Plateaued functions as an extension of bent functions play a significant role in cryptography, coding theory, sequences and combinatorics. In \cite{Mesnager9}, Mesnager \emph{et al.} introduced generalized plateaued functions in order to study plateaued functions in the general context of generalized $p$-ary functions. In this paper, we focus on the constructions of generalized $p$-ary $s$-plateaued functions from $V_{n}$ to $\mathbb{Z}_{p^k}$, where $V_{n}$ is an $n$-dimensional vector space over $\mathbb{F}_{p}$, $p$ is a prime, $k\geq 1$ and $n+s$ is even when $p=2$. In particular, when $k=1$, the constructions in this paper are applicable for plateaued functions. Firstly, inspired by the work of Hodžić \emph{et al}. \cite{Hodzic3} for Boolean plateaued functions, we characterize generalized plateaued functions with affine Walsh supports and provide constructions of generalized plateaued functions with (non)-affine Walsh supports by spectral method. When $p=2, k=1$, our constructions of Boolean plateaued functions with (non)-affine Walsh supports provide an answer to the Open Problem 2 proposed in \cite{Hodzic3}. Secondly, based on what we called generalized indirect sum, we give constructions of generalized plateaued functions, which are also applicable for (non)-weakly regular generalized bent functions. In the end, we discuss the constructions of pairwise disjoint spectra generalized plateaued functions with (non)-affine Walsh supports and we present a construction of generalized bent functions by using pairwise disjoint spectra generalized plateaued functions as building blocks.