Related papers: On $q$-nearly bent Boolean functions
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…
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…
Let $G$ be a finite abelian group acting faithfully on a finite set $X$. As a natural generalization of the perfect nonlinearity of Boolean functions, the $G$-bentness and $G$-perfect nonlinearity of functions on $X$ are studied by Poinsot…
A one to one correspondence between regular generalized bent functions from $\F_2^n$ to $\Z_{2^m},$ and $m-$tuples of Boolean bent functions is established. This correspondence maps self-dual (resp. anti-self-dual) generalized bent…
This paper introduces a novel quantum algorithm that is able to classify a hierarchy of classes of imbalanced Boolean functions. The fundamental characteristic of imbalanced Boolean functions is that the proportion of elements in their…
Given a sequence of Boolean functions $(f_n)_{n \geq 1}$, $f_n \colon \{ 0,1 \}^{n} \to \{ 0,1 \}$, and a sequence $(X^{(n)})_{n\geq 1} $ of continuous time $p_n $-biased random walks $ X^{(n)} = (X_t^{(n)})_{t \geq 0}$ on $ \{ 0,1 \}^{n}$,…
In solving $q$-difference equations, and in the definition of $q$-special functions, we encounter formal power series in which the $n$th coefficient is of size $q^{-\binom{n}{2}}$ with $q\in(0,1)$ fixed. To make sense of these formal…
In difference to many recent articles that deal with generalized bent (gbent) functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$ for certain small valued $q\in \{4,8,16 \}$, we give a complete description of these functions for both $n$…
The model of relative-error property testing of Boolean functions has been the subject of significant recent research effort [CDH+24][CPPS25a][CPPS25b] In this paper we consider the problem of relative-error testing an unknown and arbitrary…
Zhou 2013 introduced modified planar functions to describe $(2^n,2^n,2^n,1)$ relative difference sets $R$ as a graph of a function on the finite field $\F_{2^n}$, and pointed out that projections of $R$ are difference sets that can be…
It is widely claimed that the quantile function is equivariant under increasing transformations. We show by a counterexample that this is not true (even for strictly increasing transformations). However, we show that the quantile function…
A Boolean function $f:\{0,1\}^n \to \{0,1\}$ is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm showed that if the sum of…
The purpose of this paper is to detail the article of Carlet. Along the way I recall some interesting results in the theory of finite fields, I give (new) proofs of some known results, and then I generalize the construction of a family of…
Given Boolean functions \( f, g : \mathbb{F}_2^n \to \{-1,+1\} \), we say they are {\em linearly isomorphic} if there exists \( A \in \mathrm{GL}_n(\mathbb{F}_2) \) such that \( f(x)=g(Ax) \) for all \( x \). We study this problem in the…
We present a quantum algorithm for approximating the linear structures of a Boolean function $f$. Different from previous algorithms (such as Simon's and Shor's algorithms) which rely on restrictions on the Boolean function, our algorithm…
Let $q=2^e$ with $e$ even, and let $\mathbb{F}_{q^2}$ be the finite field of order $q^2$. Put $d=(q^2+q+1)/3$, and consider the permutation polynomial $$\sigma(X)=X+X^d+X^{dq}\in\mathbb{F}_{q^2}[X].$$ For $\alpha\in\mathbb{F}_q^*$, define…
Bent functions can be classified into regular bent functions, weakly regular but not regular bent functions, and non-weakly regular bent functions. Regular and weakly regular bent functions always appear in pairs since their duals are also…
A $q$-analogue of the tau function of the modified KP hierarchy is defined by a change of independent variables. This tau function satisfies a system of bilinear $q$-difference equations. These bilinear equations are translated to the…
Newton's method is an ubiquitous tool to solve equations, both in the archimedean and non-archimedean settings -- for which it does not really differ. Broyden was the instigator of what is called "quasi-Newton methods". These methods use an…
We characterize of the $q$-Bernstein functions in terms of $q$-Laplace transform. Moreover, we present several results of $q$-completely monotonic, $q$-log completely monotonic and $q$-Bernstein functions.