Constructions and Characterizations of $s$-Plateaued Partitions

2026-06-26

AI Summary

This work aims to generalize bent partitions to construct a broader class of $s$-plateaued functions and elucidate their structural properties. To this end, the notion of $s$-plateaued partitions is introduced for the first time, providing a unified framework—via vector space partitions—for constructing $p$-ary $s$-plateaued functions, vectorial functions, and generalized Boolean functions, with bent partitions emerging as a special case. By leveraging preimage structure analysis over finite fields, symmetry conditions (e.g., $-A_i = A_i$), and the theory of $(p-1)$-ary functions, the study establishes explicit constructions, cardinality estimates, and characterizations in symmetric settings. Key contributions include the successful construction of $s$-plateaued functions devoid of nonzero linear structures, a proof that under certain conditions the preimage partition itself forms an $s$-plateaued partition, and an enhanced understanding of the origins of bent partitions of depth $p^{n/2}$.

Abstract

Bent partitions play a significant role in constructing bent functions and have rich connections with coding theory and combinatorics. In this paper, we introduce $s$-plateaued partitions, which generalize the bent partitions. Let $Γ=\{A_{i}, 1 \leq i \leq K\}$ be a partition of $V_{n}^{(p)}$, where $V_{n}^{(p)}$ is an $n$-dimensional vector space over the prime field $\mathbb{F}_{p}$ and $p \mid K$. Then $Γ$ is called an $s$-plateaued partition of $V_{n}^{(p)}$ of depth $K$ if each $p$-ary function $f: V_{n}^{(p)} \rightarrow \mathbb{F}_{p}$ for which every $j \in \mathbb{F}_{p}$ has exactly $\frac{K}{p}$ of sets $A_{i}$ in $Γ$ in its preimage set, is a $p$-ary $s$-plateaued function. By using an $s$-plateaued partition, a large number of $p$-ary $s$-plateaued functions, vectorial $s$-plateaued functions and generalized $s$-plateaued functions can be constructed. In particular, $0$-plateaued partitions are just bent partitions. In general, $s$-plateaued partitions are much more complicated than bent partitions. We analyze the possible cardinality of $A_{i}$ of an $s$-plateaued partition. We give some explicit constructions of $s$-plateaued partitions for which any generated $p$-ary $s$-plateaued function has no nonzero linear structure. We give a characterization of an $s$-plateaued partition $Γ=\{A_{i}, 1 \leq i \leq K\}$, where $p$ is odd, $K \geq 5$ and $-A_{i}=A_{i}, 1 \leq i \leq K$. Based on which, we show that if $p \geq 5$, then the preimage set partition of a $p$-ary $s$-plateaued function $f: V_{n}^{(p)} \rightarrow \mathbb{F}_{p}$ with $f(x)=f(-x)$ is an $s$-plateaued partition if and only if $f$ is of $(p-1)$-form, where $n+s$ is even.When $s=0$, we partially address an open problem on whether a bent partition $Γ$ of $V_{n}^{(p)}$ of depth $p^{\frac{n}{2}}$ must be obtained from spreads.

Problem

• s-plateaued partition
• bent partition
• p-ary function
• preimage set
• linear structure

Innovation

• s-plateaued partition
• bent partition
• p-ary plateaued function
• linear structure
• (p-1)-form