Related papers: Constructing vectorial bent functions via second-o…

For any positive integers $n=2k$ and $m$ such that $m\geq k$, in this paper we show the maximal number of bent components of any $(n,m)$-functions is equal to $2^{m}-2^{m-k}$, and for those attaining the equality, their algebraic degree is…

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…

It is known that partial spreads is a class of bent partitions. In \cite{AM2022Be,MP2021Be}, two classes of bent partitions whose forms are similar to partial spreads were presented. In \cite{AKM2022Ge}, more bent partitions $\Gamma_{1},…

Generalized bent (gbent) functions is a class of functions $f: \mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$, where $q \geq 2$ is a positive integer, that generalizes a concept of classical bent functions through their co-domain extension. A lot…

Bent functions as optimal combinatorial objects are difficult to characterize and construct. In the literature, bent idempotents are a special class of bent functions and few constructions have been presented, which are restricted by the…

Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group $(\gf(2^{2m}), +)$, have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property. The main objective…

In this article, we study bent functions on $\mathbb{F}_2^{2m}$ of the form $f(x,y) = x \cdot \phi(y) + h(y)$, where $x \in \mathbb{F}_2^{m-1} $ and $ y \in \mathbb{F}_2^{m+1}$, which form the generalized Maiorana-McFarland class (denoted…

We study vectorial functions with maximal number of bent components in this paper. We first study the Walsh transform and nonlinearity of $F(x)=x^{2^e}h(\Tr_{2^{2m}/2^m}(x))$, where $e\geq0$ and $h(x)$ is a permutation over $\F_{2^m}$. If…

There are two construction methods of designs from $(n,m)$-bent functions, known as translation and addition designs. In this paper we analyze, which equivalence relation for Boolean bent functions, i.e. $(n,1)$-bent functions, and…

In this article a procedure to construct bent functions from $\F_{p^n}$ to $\F_p$ by merging plateaued functions which are bent on ($n-2$)-dimensional subspaces of $\F_{p^n}$ is presented. Taking advantage of such classes of plateaued…

In \cite{Bapic, Tang, Zheng} a new method for the secondary construction of vectorial/Boolean bent functions via the so-called $(P_U)$ property was introduced. In 2018, Qi et al. generalized the methods in \cite{Tang} for the construction…

In this work, we employ the concept of {\em composite representation} of Boolean functions, which represents an arbitrary Boolean function as a composition of one Boolean function and one vectorial function, for the purpose of specifying…

In 2017, Zhang et al. proposed a question (not open problem) and two open problems in [IEEE TIT 63 (8): 5336--5349, 2017] about constructing bent functions by using Rothaus' construction. In this note, we prove that the sufficient…

In this paper, we first present a new secondary construction of bent functions (building new bent functions from two already defined ones). Furthermore, we apply the construction using as initial functions some specific bent functions and…

In this paper, we consider the characterization of the bentness of quadratic Boolean functions of the form $f(x)=\sum_{i=1}^{\frac{m}{2}-1} Tr^n_1(c_ix^{1+2^{ei}})+ Tr_1^{n/2}(c_{m/2}x^{1+2^{n/2}}) ,$ where $n=me$, $m$ is even and $c_i\in…

Bent functions are balanced by restricting their domains to vectors with either even or odd Hamming weights, which ensures an equal number of pre-images for both, 0 and 1. Using the previous fact, we can construct bent functions on two…

This note contains a simple construction of complete sets of MUBs, using bent functions to write the new basis vectors as explicit linear combinations of the standard basis.

Let $\mathbb{F}_{p^{n}}$ be the finite field with $p^n$ elements and $\operatorname{Tr}(\cdot)$ be the trace function from $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_{p}$, where $p$ is a prime and $n$ is an integer. Inspired by the works of…

Bent functions, which are maximally nonlinear Boolean functions with even numbers of variables and whose Hamming distance to the set of all affine functions equals $2^{n-1}\pm 2^{\frac{n}{2}-1}$, were introduced by Rothaus in 1976 when he…

Bent functions from a vector space $V_n$ over $\mathbb F_2$ of even dimension $n=2m$ into the cyclic group $\mathbb Z_{2^k}$, or equivalently, relative difference sets in $V_n\times\mathbb Z_{2^k}$ with forbidden subgroup $\mathbb Z_{2^k}$,…