constructions can also provide new plateaued vectorial functions.'
author:
- 'Lijing Zheng $^{\diamond}$, Jie Peng $^{*}$, Haibin Kan $^{\diamond }$, Yanjun Li$~^{*}$, and Juan Luo [^1]'
title: 'On constructions and properties of $(n,m)$-functions with maximal number of bent components'
---

[**Index Terms:**]{} Bent functions, vectorial bent, algebraic degree, differential uniformity, Walsh spectrum.

[**MSC:**]{} 94A60, 11T71, 14G50.

Introduction
============

Throughout this paper, we often identify the finite field $\mathbb{F}_{2^{n}}$ with $\mathbb{F}^{n}_{2}$, the $n$-dimensional vector space over $\mathbb{F}_{2}$. Any function $F: \mathbb{F}_{2^{n}}\rightarrow\mathbb{F}_{2^{m}}$ is called an [*$(n,m)$-function*]{}. When $m=1$ it is usually called a [*Boolean function*]{}. [*Bent functions*]{}, as a special class of Boolean functions, were introduced by Rothaus [@Rothaus] in 1976. A bent function is a Boolean function in even number of variables which is maximally nonlinear in the sense that its Hamming distance to all affine functions is optimal. It corresponds to the fact that the [*Walsh transform*]{} of a bent function in $n$ variables takes precisely the values $\pm 2^{\frac{n}{2}}$. Over the last four decades, bent functions have attracted a lot of research interest because of their relations to coding theory and applications in cryptography. A survey on bent functions can be found in [@Car-Mes16] as well as the book 