A Further Study of Vectorial Dual-Bent Functions

TL;DR

This paper explores new characterizations and constructions of vectorial dual-bent functions, revealing their connections to association schemes, linear codes, and Hadamard matrices, and extends understanding of bent partitions for cryptographic applications.

Contribution

It introduces new characterizations of vectorial dual-bent functions with Condition A and provides necessary and sufficient conditions for constructing association schemes from these functions.

Findings

Characterizations of vectorial dual-bent functions with Condition A in terms of association schemes, codes, and Hadamard matrices.

Necessary and sufficient conditions for constructing association schemes from vectorial dual-bent functions.

New methods to construct association schemes from vectorial dual-bent functions.

Abstract

Vectorial dual-bent functions have recently attracted some researchers' interest as they play a significant role in constructing partial difference sets, association schemes, bent partitions and linear codes. In this paper, we further study vectorial dual-bent functions $F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}$, where $2\leq m \leq \frac{n}{2}$, $V_{n}^{(p)}$ denotes an $n$-dimensional vector space over the prime field $\mathbb{F}_{p}$. We give new characterizations of certain vectorial dual-bent functions (called vectorial dual-bent functions with Condition A) in terms of amorphic association schemes, linear codes and generalized Hadamard matrices, respectively. When $p=2$, we characterize vectorial dual-bent functions with Condition A in terms of bent partitions. Furthermore, we characterize certain bent partitions in terms of amorphic association schemes, linear codes and generalized Hadamard matrices, respectively. For general vectorial dual-bent functions $F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}$ with $F(0)=0, F(x)=F(-x)$ and $2\leq m \leq \frac{n}{2}$, we give a necessary and sufficient condition on constructing association schemes. Based on such a result, more association schemes are constructed from vectorial dual-bent functions.