Related papers: On quadratic binomial vectorial functions with max…

Using recent results on solving the equation $X^{2^k+1}+X+a=0$ over a finite field $\mathbb{F}_{2^n}$, we address an open question raised by the first author in WAIFI 2014 concerning the APN-ness of the Kasami functions $x\mapsto…

The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P =…

A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in R^n extends to a monomial transformation of R^n that preserves w. MacWilliams proved that finite fields with the…

We prove a generalization of the Hermitian version of the Helton-Vinnikov determinantal representation of hyperbolic polynomials to the class of semi-hyperbolic polynomials, a strictly larger class, as shown by an example. We also prove…

We obtain the following results. For any prime $q$ the minimal Hamming distance between distinct regular $q$-ary bent functions of $2n$ variables is equal to $q^n$. The number of $q$-ary regular bent functions at the distance $q^n$ from the…

We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose…

The necessary and sufficient conditions for a class of functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$, where $q \geq 2$ is an even positive integer, have been recently identified for $q=4$ and $q=8$. In this article we give an…

In the light of a series of papers on moving vectors, we define and study core abaci of classical affine types for arbitrary charge. This greatly extends the concept of cores with charge zero, and make us being able to parameterize the…

An eight-component formalism is proposed for the relativistic two-fermion problem. In QED, it extends the applicability of the Dirac equation with hyperfine interaction to the positronium case. The use of exact relativistic two-body…

Pickands dependence functions characterize bivariate extreme value copulas. In this paper, we study the class of polynomial Pickands functions. We provide a solution for the characterization of such polynomials of degree at most $m+2$,…

In this paper we generalize the partial spread class and completely describe it for generalized Boolean functions from $\F_2^n$ to $\mathbb{Z}_{2^t}$. Explicitly, we describe gbent functions from $\F_2^n$ to $\mathbb{Z}_{2^t}$, which can be…

We show that a fractal affine function $f(x)$ defined by a system $\mathcal S$ which does not satisfy weak separation property is a quadratic function.

Suppose that A is a subset of F_2^n of density as close to 1/3 as possible. We show that the A(F_2^n)-norm (that is the sum of the absolute values of the Fourier transform) of the characterstic function of A is bounded below by an absolute…

We investigate transitive uniform partitions of the vector space $F^n$ of dimension $n$ over the Galois field $GF(2)$ into cosets of Hamming codes. A partition $P^n= \{H_0,H_1+e_1,\ldots,H_n+e_n\}$ of $F^n$ into cosets of Hamming codes…

We consider a family of generic weighted arrangements of $n$ hyperplanes in $\C^k$ and show that the Gauss-Manin connection for the associated hypergeometric integrals, the contravariant form on the space of singular vectors, and the…

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…

We give a sufficient condition for a bi-invariant weight on a Frobenius bimodule to satisfy the extension property. This condition applies to bi-invariant weights on a finite Frobenius ring as a special case. The complex-valued functions on…

Consider an absolutely irreducible polynomial $F(Y,X_1,\ldots,X_n) \in \mathbb{Z}[Y,X_1,\ldots,X_n]$ that is monic in $Y$ and is a polynomial in $Y^m$ for an integer $m \geq 1$. Let $N(F,B)$ count the number of $\mathbf{x} \in [-B,B]^n \cap…

Feistel Boomerang Connectivity Table (FBCT) is an important cryptanalytic technique on analysing the resistance of the Feistel network-based ciphers to power attacks such as differential and boomerang attacks. Moreover, the coefficients of…

We define generalized Hamming weights for almost affine codes. We show how various aspects and applications of generalized Hamming weights for linear codes, such as Wei duality, generalized Kung's bound, profiles, connection to wire-tap…