Related papers: On q-ary Bent and Plateaued Functions

The minimal Hamming distance between distinct $p$-ary bent functions of $2n$ variables is proved to be $p^n$ for any prime $p$. It is shown that the number of $p$-ary bent functions at the distance $p^n$ from the quadratic bent function is…

For each non-constant $q$ in the set of $n$-variable Boolean functions, the {\em $q$-transform} of a Boolean function $f$ is related to the Hamming distances from $f$ to the functions obtainable from $q$ by nonsingular linear change of…

The logarithm of the number of binary n-variable bent functions is asymptotically less than $11(2^n)/32$ as n tends to infinity. Keywords: boolean function, Walsh--Hadamard transform, plateaued function, bent function, upper bound

We show that the graph of a bent function is a Salem set in an appropriate sense. We also establish a simple result that quantifies redundancies in the difference operators of a function, which applies to bent functions over fields of odd…

For each non-constant Boolean function $q$, Klapper introduced the notion of $q$-transforms of Boolean functions. The {\em $q$-transform} of a Boolean function $f$ is related to the Hamming distances from $f$ to the functions obtainable…

The number of $n$-ary bent functions is less than $2^{3\cdot2^{n-3}(1+o(1))}$ as $n$ is even and $n\rightarrow\infty$. Keywords: Boolean function, bent function, upper bound

A function $F:\mathbb{F}_2^n\rightarrow \mathbb{F}_2^n$, $n=2m$, can have at most $2^n-2^m$ bent component functions. Trivial examples are obtained as $F(x) = (f_1(x),\ldots,f_m(x),a_1(x),\ldots, a_m(x))$, where…

Bent functions of the form $\mathbb{F}_2^n\rightarrow\mathbb{Z}_q$, where $q\geqslant2$ is a positive integer, are known as generalized bent (gbent) functions. Gbent functions for which it is possible to define a dual gbent function are…

We study codes with parameters of $q$-ary shortened Hamming codes, i.e., $(n=(q^m-q)/(q-1), q^{n-m}, 3)_q$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold…

Let $\mathbb{F}_q$ be the finite field of order $q$ and $E\subset \mathbb{F}_q^d$, where $4|d$. Using Fourier analytic techniques, we prove that if $|E|>\frac{q^{d-1}}{d}\binom{d}{d/2}\binom{d/2}{d/4}$, then the points of $E$ determine a…

In this article a technique for constructing $p$-ary bent functions from near-bent functions is presented. Two classes of quadratic $p$-ary functions are shown to be near-bent. Applying the construction of bent functions to these classes of…

In this paper, we study the Hamming distance between vectorial Boolean functions and affine functions. This parameter is known to be related to the non-linearity and differential uniformity of vectorial functions, while the calculation of…

We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from $\mathbb{F}_q^n$ to $\mathbb{F}_q$ to the set of affine functions from $\mathbb{F}_q^n$ to $\mathbb{F}_q$. We prove…

Whether two distinct APN functions can have a Hamming distance of $1$ remains an open problem. In 2020, L. Budaghyan et al. introduced a new CCZ-invariant $\Pi_F$ which can be used to provide lower bounds on the Hamming distance between a…

We give a new simple construction for known binary quadratic symmetric bent and almost bent functions. In particular, for even number of variables, they are self-dual and anti-self-dual quadratic bent functions, respectively, which are not…

In this paper, we describe the properties of the $i$-components of Hamming codes. We suggest constructions of the admissible families of components of Hamming codes. It is shown that every $q$-ary code of length $m$ and minimum distance 5…

In this paper, we investigate properties of functions from $\mathbb{Z}_{p}^n$ to $\mathbb{Z}_q$, where $p$ is an odd prime and $q$ is a positive integer divided by $p$. we present the sufficient and necessary conditions for bent-ness 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…

For any finite field ${\mathbb F}_q$ with $q$ elements, we study the set ${\mathcal F}_{(q,m)}$ of functions from ${\mathbb F}_q^m$ into ${\mathbb F}^q$. We introduce a transformation that allows us to determine a linear system of $q^{m+1}$…

The Hamming graph $H(n,q)$ is the graph whose vertices are the words of length $n$ over the alphabet $\{0,1,\ldots,q-1\}$, where two vertices are adjacent if they differ in exactly one coordinate. The adjacency matrix of $H(n,q)$ has $n+1$…