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Daniele Bartoli

Publications and source records attributed to Daniele Bartoli.

At least 73 records · Page 4Linked to original sources
arXiv2019

New examples of maximal curves with low genus

We investigate the Jacobian decomposition of some algebraic curves over finite fields with genus $4$, $5$ and $10$. As a corollary, explicit equations for curves that are either maximal or minimal over the finite field with $p^2$ elements are obtained for infinitely many $p$'s. Lists of small $p$'s for which maximality holds are provided. In some cases we describe the automorphism group of the curve.

math.AG↗
arXiv2019

A family of permutation trinomials in $\mathbb{F}_{q^2}$

Let $p>3$ and consider a prime power $q=p^h$. We completely characterize permutation polynomials of $\mathbb{F}_{q^2}$ of the type $f_{a,b}(X) = X(1 + aX^{q(q-1)} + bX^{2(q-1)}) \in \mathbb{F}_{q^2}[X]$. In particular, using connections with algebraic curves over finite fields, we show that the already known sufficient conditions are also necessary.

math.CO↗
arXiv2019

The Diophantine Equation $(x+1)^k+(x+2)^k+\cdots+(\ell x)^k=y^n$ Revisited

Let $k,\ell\geq2$ be fixed integers and $C$ be an effectively computable constant depending only on $k$ and $\ell$. In this paper, we prove that all solutions of the equation $(x+1)^{k}+(x+2)^{k}+...+(\ell x)^{k}=y^{n}$ in integers $x,y,n$ with $x,y\geq1, n\geq2, k\neq3$ and $\ell\equiv 1 \pmod 2$ satisfy $\max\{x,y,n\}<C$. The case when $\ell$ is even has already been completed by Soydan (Publ. Math. Debrecen 91 (2017), pp. 369-382).

math.NT↗
arXiv2019

Bent functions from triples of permutation polynomials

We provide constructions of bent functions using triples of permutations. This approach is due to Mesnager. In general, involutions have been mostly considered in such a machinery; we provide some other suitable triples of permutations, using monomials, binomials, trinomials, and quadrinomials.

math.CO↗
arXiv2019

Towards the full classification of exceptional scattered polynomials

Let $f(X) \in \mathbb{F}_{q^r}[X]$ be a $q$-polynomial. If the $\mathbb{F}_q$-subspace $U=\{(x^{q^t},f(x)) \mid x \in \mathbb{F}_{q^n}\}$ defines a maximum scattered linear set, then we call $f(X)$ a scattered polynomial of index $t$. The asymptotic behaviour of scattered polynomials of index $t$ is an interesting open problem. In this sense, exceptional scattered polynomials of index $t$ are those for which $U$ is a maximum scattered linear set in ${\rm PG}(1,q^{mr})$ for infinitely many $m$. The complete classifications of exceptional scattered monic polynomials of index $0$ (for $q>5$) and of index 1 were obtained by Bartoli and Zhou. In this paper we complete the classifications of exceptional scattered monic polynomials of index $0$ for $q \leq 4$. Also, some partial classifications are obtained for arbitrary $t$. As a consequence, the complete classification of exceptional scattered monic polynomials of index $2$ is given.

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arXiv2018

Weierstrass semigroups at every point of the Suzuki curve

In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Suzuki curve $\mathcal{S}_q$. As the point $P$ varies, exactly two possibilities arise for $H(P)$: one for the $\mathbb{F}_q$-rational points (already known in the literature), and one for all remaining points. For this last case a minimal set of generators of $H(P)$ is also provided. As an application, we construct dual one-point codes from an $\mathbb{F}_{q^4}\setminus\fq$-point whose parameters are better in some cases than the ones constructed in a similar way from an $\fq$-rational point.

math.CO↗
arXiv2018

Minimal linear codes in odd characteristic

In this paper we generalize constructions in two recent works of Ding, Heng, Zhou to any field $\mathbb{F}_q$, $q$ odd, providing infinite families of minimal codes for which the Ashikhmin-Barg bound does not hold.

math.CO↗
arXiv2018

Low-degree planar polynomials over finite fields of characteristic two

Planar functions are mappings from a finite field $\mathbb{F}_q$ to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of $q$ and we consider the case that $q$ is even. We classify polynomials of degree at most $q^{1/4}$ that induce planar functions on $\mathbb{F}_q$, by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over $\mathbb{F}_q$ that induce planar functions on infinitely many extensions of~$\mathbb{F}_q$. The proof strategy is to study the number of $\mathbb{F}_q$-rational points of an algebraic curve attached to a putative planar function.~Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear~functions.

math.CO↗
arXiv2018

Permutation polynomials and complete permutation polynomials over $\mathbb{F}_{q^{3}}$

Motivated by many recent constructions of permutation polynomials over $\mathbb{F}_{q^2}$, we study permutation polynomials over $\mathbb{F}_{q^3}$ in terms of their coefficients. Based on the multivariate method and resultant elimination, we construct several new classes of sparse permutation polynomials over $\mathbb{F}_{q^3}$, $q=p^{k}$, $p\geq3$. Some of them are complete mappings.

math.NT↗
arXiv2018

Permutation trinomials over $\mathbb{F}_{q^3}$

We consider four classes of polynomials over the fields $\mathbb{F}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+Ax^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+Ax^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+Ax^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+Ax^{q}-Bx$, where $A,B \in \mathbb{F}_q$. We determine conditions on the pairs $(A,B)$ and we give lower bounds on the number of pairs $(A,B)$ for which these polynomials permute $\mathbb{F}_{q^3}$.

math.CO↗
arXiv2018

Pure gaps on curves with many rational places

We consider the algebraic curve defined by $y^m = f(x)$ where $m \geq 2$ and $f(x)$ is a rational function over $\mathbb{F}_q$. We extend the concept of pure gap to {\bf c}-gap and obtain a criterion to decide when an $s$-tuple is a {\bf c}-gap at $s$ rational places on the curve. As an application, we obtain many families of pure gaps at two rational places on curves with many rational places.

math.CO↗
arXiv2018

Permutation polynomials over $\mathbb{F}_{q^2}$ from rational functions

Let $μ_{q+1}$ denote the set of $(q+1)$-th roots of unity in $\mathbb{F}_{q^2 }$. We construct permutation polynomials over $\mathbb{F}_{q^2}$ by using rational functions of any degree that induce bijections either on $μ_{q+1}$ or between $μ_{q+1}$ and $\mathbb{F}_q \cup \{\infty\}$. In particular, we generalize results from Zieve.

math.CO↗
arXiv2018

$\mathbb{F}_{p^2}$-maximal curves with many automorphisms are Galois-covered by the Hermitian curve

Let $\mathbb{F}$ be the finite field of order $q^2$, $q=p^h$ with $p$ prime. It is commonly atribute to J.P. Serre the fact that any curve $\mathbb{F}$-covered by the Hermitian curve $\mathcal{H}_{q+1}:\, y^{q+1}=x^q+x$ is also $\mathbb{F}$-maximal. Nevertheless, the converse is not true as the Giulietti-Korchmáros example shows provided that $q>8$ and $h\equiv 0\pmod{3}$. In this paper, we show that if an $\mathbb{F}$-maximal curve $\mathcal{X}$ of genus $g\geq 2$ where $q=p$ is such that $|Aut(\mathcal{X})|>84(g-1)$ then $\mathcal{X}$ is Galois-covered by $\mathcal{H}_{p+1}$. Also, we show that the hypothesis on the order of $Aut(\mathcal{X})$ is sharp, since there exists an $\mathbb{F}$-maximal curve $\mathcal{X}$ for $q=71$ of genus $g=7$ with $|Aut(\mathcal{X})|=84(7-1)$ which is not Galois-covered by the Hermitian curve $\mathcal{H}_{72}$.

math.AG↗
arXiv2018

Tables, bounds and graphics of sizes of complete arcs in the plane $\mathrm{PG}(2,q)$ for all $q\le321007$ and sporadic $q$ in $[323761\ldots430007]$ obtained by an algorithm with fixed order of points (FOP)

In the previous works of the authors, a step-by-step algorithm FOP which uses any fixed order of points in the projective plane $\mathrm{PG}(2,q)$ is proposed to construct small complete arcs. In each step, the algorithm adds to a current arc the first point in the fixed order not lying on the bisecants of the arc. The algorithm is based on the intuitive postulate that $\mathrm{PG}(2,q)$ contains a sufficient number of relatively small complete arcs. Also, in the previous papers, it is shown that the type of order on the points of $\mathrm{PG}(2,q)$ is not relevant. A complete lexiarc in $\mathrm{PG}(2,q)$ is a complete arc obtained by the algorithm FOP using the lexicographical order of points. In this work, we collect and analyze the sizes of complete lexiarcs in the following regions: \begin{align*}& \textbf{all } q\le321007,~ q \mbox{ prime power}; & 15 \mbox{ sporadic $q$'s in the interval }[323761\ldots430007], \mbox{ see (1.10)}. \end{align*} In the work [9], the smallest known sizes of complete arcs in $\mathrm{PG}(2,q)$ are collected for all $q\leq160001$, $q$ prime power. The sizes of complete arcs, collected in this work and in [9], provide the following upper bounds on the smallest size $t_{2}(2,q)$ of a complete arc in the projective plane $\mathrm{PG}(2,q)$: \begin{align*} t_{2}(2,q)&<0.998\sqrt{3q\ln q}<1.729\sqrt{q\ln q}&\mbox{ for }&&7&\le q\le160001;\\ t_{2}(2,q)&<1.05\sqrt{3q\ln q}<1.819\sqrt{q\ln q}&\mbox{ for }&&7&\le q\le321007. \end{align*} Our investigations and results allow to conjecture that the bound $t_{2}(2,q)<1.05\sqrt{3q\ln q}<1.819\sqrt{q\ln q}$ holds for all $q\ge7$. It is noted that sizes of the random complete arcs and complete lexiarcs behave similarly. This work can be considered as a continuation and development of the paper [11].

math.CO↗
arXiv2017

On a conjecture about a class of permutation trinomials

We prove a conjecture by Tu, Zeng, Li, and Helleseth concerning trinomials $f_{α,β}(x)= x + αx^{q(q-1)+1} + βx^{2(q-1)+1} \in \mathbb{F}_{q^2}[x]$, $αβ\neq 0$, $q$ even, characterizing all the pairs $(α,β)\in \mathbb{F}_{q^2}^2$ for which $f_{α,β}(x)$ is a permutation of $\mathbb{F}_{q^2}$.

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