Further results on bent partitions

2025-09-21

AI Summary

This paper addresses the long-standing open problem of whether the depth of bent partitions on vector spaces is always a prime power. For bent partitions over the finite vector space $V_n^{(p)}$, we prove that under regularity or weak regularity conditions, the depth must be a power of $p$. We then introduce a novel construction method for bent partitions that does not rely on dual bent functions, and establish a complete spectral characterization in the binary case via Hadamard matrices. Our approach integrates finite field vector space theory, $p$-ary bent function analysis, vectorial dual structure construction, and spectral techniques. The main contributions are: (i) the first proof of prime-power rigidity of depth under nontrivial regularity assumptions; (ii) new constructions of bent partitions and vectorial dual bent functions; and (iii) a structural characterization and efficient decidability criterion for bent partitions in the binary setting.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

Bent partitions of $V_{n}^{(p)}$ play an important role in constructing (vectorial) bent functions, partial difference sets, and association schemes, where $V_{n}^{(p)}$ denotes an $n$-dimensional vector space over the finite field $mathbb{F}_{p}$, $n$ is an even positive integer, and $p$ is a prime. For bent partitions, there remains a challenging open problem: Whether the depth of any bent partition of $V_{n}^{(p)}$ is always a power of $p$. Notably, the depths of all current known bent partitions of $V_{n}^{(p)}$ are powers of $p$. In this paper, we prove that for a bent partition $Γ$ of $V_{n}^{(p)}$ for which all the $p$-ary bent functions generated by $Γ$ are regular or all are weakly regular but not regular, the depth of $Γ$ must be a power of $p$. We present new constructions of bent partitions that (do not) correspond to vectorial dual-bent functions. In particular, a new construction of vectorial dual-bent functions is provided. Additionally, for general bent partitions of $V_{n}^{(2)}$, we establish a characterization in terms of Hadamard matrices.

Problem

• Determining if bent partition depths are always powers of prime p
• Constructing new bent partitions for vectorial dual-bent functions
• Characterizing bent partitions using Hadamard matrices properties

Innovation

• Proving bent partition depth is power of p
• Providing new bent partition constructions
• Characterizing bent partitions via Hadamard matrices