On the maximum number of rational points on singular curves over finite fields

We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus $\pi$

Differentially 4-uniform functions

We give a geometric characterization of vectorial boolean functions with differential uniformity less or equal to 4

On a conjecture of Helleseth

We are concern about a conjecture proposed in the middle of the seventies by Hellesseth in the framework of maximal sequences and theirs cross-correlations. The conjecture claims the existence of a zero outphase Fourier coefficient. We give some divisibility properties in this direction

Extended core and choosability of a graph

A graph $G$ is $(a,b)$-choosable if for any color list of size $a$ associated with each vertices, one can choose a subset of $b$ colors such that adjacent vertices are colored with disjoint color sets. This paper shows an equivalence between the $(a,b)$-choosability of a graph and the $(a,b)$-choosability of one of its subgraphs called the extended core. As an application, this result allows to prove the $(5,2)$-choosability and $(7,3)$-colorability of triangle-free induced subgraphs of the triangular lattice.Comment: 10 page

Class number in non Galois quartic and non abelian Galois octic function fields over finite fields

International audienceWe consider a totally imaginary extension of a real extension of a rational function field over a finite field of odd characteristic. We prove that the relative ideal class number one problem for such non Galois quartic fields is equivalent to the one for non abelian Galois octic imaginary functions fields. Then, we develop some results on characters which give a method to evaluate the ideal class number of such quartic function fields

Maximal differential uniformity polynomials

We provide an explicit infinite family of integers $m$ such that all the polynomials of ${\mathbb F}_{2^n}[x]$ of degree $m$ have maximal differential uniformity for $n$ large enough. We also prove a conjecture of the third author in these cases

Choosability of a weighted path and free-choosability of a cycle

A graph $G$ with a list of colors $L(v)$ and weight $w(v)$ for each vertex $v$ is $(L,w)$-colorable if one can choose a subset of $w(v)$ colors from $L(v)$ for each vertex $v$, such that adjacent vertices receive disjoint color sets. In this paper, we give necessary and sufficient conditions for a weighted path to be $(L,w)$-colorable for some list assignments $L$. Furthermore, we solve the problem of the free-choosability of a cycle.Comment: 9 page