arXiv subjects

Marco Timpanella

Publications and source records attributed to Marco Timpanella.

At least 19 recordsLinked to original sources

arXiv2026

Curves with a large automorphism group admitting a cyclic subgroup of index $2$

The Hurwitz bound on the order of the $\mathbb K$-automorphism group ${\rm{Aut}}({\mathcal{X}})$ of an algebraic curve ${\mathcal{X}}$ of genus $g(\mathcal{X})\ge 2$ defined over a field $\mathbb K$ of zero characteristic states that $|{\rm{Aut}}({\mathcal{X}})|\le 84(g(\mathcal{X})-1)$. Improved bounds are available for the order of certain types of subgroups within automorphism groups. For instance, if a subgroup $H$ of ${\rm{Aut}}({\mathcal{X}})$ is dihedral, then in the complex case, $|H| \leq 4g(\mathcal{X}) + 4$. More recently it has been shown that a tighter bound holds for $H$ a generalized quasi-dihedral group. In this paper we explore the more general setting of a curve defined over a field of any characteristic, and $H$ a group admitting a cyclic subgroup of index two. We show that the same upper bound for the size of a dihedral group of automorphisms holds for curves defined over an algebraically closed field of characteristic $p\ne 2$. Then we provide some classification results about (non-dihedral) groups of size larger than $4g(\mathcal{X})+4$ admitting a cyclic subgroup of index $2$.

math.AG↗

arXiv2026

Large automorphism groups compared to the $p$-rank of algebraic curves in characteristic $p$

Let $\cX$ be a (projective, geometrically irreducible, non-singular) algebraic curve of genus $\ge 2$ and positive $p$-rank $\gamma(\cX)$, defined over an algebraically closed field $\mathbb{K}$ of positive characteristic $p>0$. Contrary to what occurs for the genera, no function $h(\gamma)$ exists such that $|\aut(\cX)|\le h(\gamma)$ whenever $\gamma=\gamma(\cX)$. Thus, to have a bound on $|\aut(\cX)|$ only depending on $\gamma(\cX)$, some restrictions on $\cX$ and $\aut(\cX)$ are needed. In this context, the following theorem is proven. Let $\Gamma$ be a subgroup of $\aut(\cX)$. Assume the existence of a point $P\in \cX$ such that if $S_P$ is the Sylow $p$-subgroup of $\Gamma_P$ fixing $P$, then the quotient curve $\cX/S_P$ is rational. Then %$\gamma(\cX)\ge 2$ and the following $p$-rank analog of the Riemann-Hurwitz bound \begin{equation*} %\label{eq18122025} |\Gamma|<900 \left(\frac{p}{p-1}\right)^4 \gamma(\cX)^4 \end{equation*} holds, unless a subgroup of index $\le 2$ of $\Gamma$ fixes $P$. This bound is sharp apart from the constant.

math.AG↗

arXiv2026

A complete characterization of a family of permutation trinomials over $\mathbb F_{p^2}$

Let $p>3$ be a prime and let $$f_{\lambda_1,\lambda_2}(x)=x^{p^2-p+1}+\lambda_1x^{p^2}+\lambda_2x^{2p-1}\in\mathbb F_{p^2}[x].$$ We determine all pairs $(\lambda_1,\lambda_2)\in(\mathbb F_{p^2})^2$ for which $f_{\lambda_1,\lambda_2}$ is a permutation polynomial of $\mathbb F_{p^2}$. The final classification consists of three explicit families. The first one is the binomial case $\lambda_1=0$. The other two are obtained from the condition $\lambda_2=c\lambda_1^3$, with $c\in \mathbb F_{p}^{*}$, and are defined by two simple equations involving the norm $\lambda_1^{p+1}$. The proof is based on the AGW criterion and on the study of a quartic curve naturally associated with the rational function induced on the unit circle $\mu_{p+1}$.

math.CO↗

arXiv2026

Generalizing a family of scattered quadrinomials in $\mathbb{F}_{q^{2t}}[X]$

In recent years, several efforts have focused on identifying new families of scattered polynomials. Currently, only three families in $\mathbb{F}_{q^n}[X]$ are known to exist for infinitely many values of $n$ and $q$: (i) pseudoregulus-type monomials, (ii) Lunardon-Polverino-type binomials, and (iii) a family of quadrinomials studied in a series of papers. In this work, we provide sufficient conditions under which these quadrinomials, denoted by $\psi_{m,h,s}$, are scattered. Our results both include and generalize those obtained in previous studies. We also investigate the equivalences between the previously known families of scattered polynomials and those in this new class.

math.CO↗

arXiv2025

AG codes from the Hermitian curve for Cross-Subspace Alignment in Private Information Retrieval

Private information retrieval (PIR) addresses the problem of retrieving a desired message from distributed databases without revealing which message is being requested. Recent works have shown that cross-subspace alignment (CSA) codes constructed from algebraic geometry (AG) codes on high-genus curves can improve PIR rates over classical constructions. In this paper, we propose a new PIR scheme based on AG codes from the Hermitian curve, a well-known example of an $F_\ell$-maximal curve, that is, a curve defined over the finite field with $\ell$ elements which attains the Hasse-Weil upper bound on the number of its $F_\ell$-rational points. The large number of rational points enables longer code constructions, leading to higher retrieval rates than schemes based on genus 0, genus 1, and hyperelliptic curves of arbitrary genus. Our results highlight the potential of maximal curves as a natural source of efficient PIR constructions.

math.AG↗

arXiv2025

Ovoids of $Q^+(7,q)$ of low-degree

Ovoids of the hyperbolic quadric $Q^+(7,q)$ of $\mathrm{PG}(7,q)$ have been extensively studied over the past 40 years, partly due to their connections with other combinatorial objects. It is well known that the points of an ovoid of $Q^+(7,q)$ can be parametrized by three polynomials $f_1(X,Y,Z)$, $f_2(X,Y,Z)$, $f_3(X,Y,Z)$. In this paper, we classify ovoids of $Q^+(7,q)$ of low degree, specifically under the assumption that $f_1(X,Y,Z)$, $f_2(X,Y,Z)$, $f_3(X,Y,Z)$ have degree at most 3. Our approach relies on the analysis of an algebraic hypersurface associated with the ovoid.

math.CO↗

arXiv2024

Algebraic curves with a large cyclic automorphism group

The study of algebraic curves $\cX$ with numerous automorphisms in relation to their genus $g(\cX)$ is a well-established area in Algebraic Geometry. In 1995, Irokawa and Sasaki \cite{Sasaki} gave a complete classification of curves over $\mathbb{C}$ with an automorphism of order $N \geq 2g(\mathcal{X}) + 1$. Precisely, such curves are either hyperelliptic with $N=2g(\cX)+2$ with $g(\cX)$ even, or are quotients of the Fermat curve of degree $N$ by a cyclic group of order $N$. Such a classification does not hold in positive characteristic $p$, the curve with equation $y^2=x^p-x$ being a well-studied counterexample. This paper successfully classifies curves with a cyclic automorphism group of order $N$ at least $2g(\mathcal{X}) + 1$ in positive characteristic $p \neq 2$, offering the positive characteristic counterpart to the Irokawa-Sasaki result. The possibility of wild ramification in positive characteristic has presented a few challenges to the investigation.

math.AG↗

arXiv2023

Scattered trinomials of $\mathbb{F}_{q^6}[X]$ in even characteristic

In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In [B. Csajb\'ok, G. Marino and F. Zullo: New maximum scattered linear sets of the projective line, Finite Fields Appl. 54 (2018), 133-150; G. Marino, M. Montanucci and F. Zullo: MRD-codes arising from the trinomial $x^q+x^{q^3}+cx^{q^5}\in\mathbb{F}_{q^6}[x]$, Linear Algebra Appl. 591 (2020), 99-114], the authors proved that the trinomial $f_c(X)=X^{q}+X^{q^{3}}+cX^{q^{5}}$ of $\mathbb{F}_{q^6}[X]$ is scattered under the assumptions that $q$ is odd and $c^2+c=1$. They also explicitly observed that this is false when $q$ is even. In this paper, we provide a different set of conditions on $c$ for which this trinomial is scattered in the case of even $q$. Using tools of algebraic geometry in positive characteristic, we show that when $q$ is even and sufficiently large, there are roughly $q^3$ elements $c \in \mathbb{F}_{q^6}$ such that $f_{c}(X)$ is scattered. Also, we prove that the corresponding MRD-codes and $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^6)$ are not equivalent to the previously known ones.

math.CO↗

arXiv2023

Two-point AG codes from one of the Skabelund maximal curves

In this paper, we investigate two-point Algebraic Geometry codes associated to the Skabelund maximal curve constructed as a cyclic cover of the Suzuki curve. In order to estimate the minimum distance of such codes, we make use of the generalized order bound introduced by P. Beelen and determine certain two-point Weierstrass semigroups of the curve.

math.AG↗

arXiv2023

New sextics of genus 6 and 10 attaining the Serre bound

We provide new examples of curves of genus 6 or 10 attaining the Serre bound. They all belong to the family of sextics introduced in [19] as a a generalization of the Wiman sextics [36] and Edge sextics [9]. Our approach is based on a theorem by Kani and Rosen which allows, under certain assumptions, to fully decompose the Jacobian of the curve. With our investigation we are able to update several entries in \url{http://www.manypoints.org} ([35]).

math.AG↗

arXiv2023

Complete $(q+1)$-arcs in $\mathrm{PG}(2,\mathbb{F}_{q^6})$ from the Hermitian curve

We prove that, if $q$ is large enough, the set of the $\mathbb{F}_{q^6}$-rational points of the Hermitian curve is a complete $(q+1)$-arc in $\mathrm{PG}(2,\mathbb{F}_{q^6})$, addressing an open case from a recent paper by Korchm\'aros, Sz\H{o}nyi and Nagy. An algebraic approach based on the investigation of some algebraic varieties attached to the arc is used.

math.CO↗

arXiv2023

On AG codes from a generalization of the Deligne-Lustzig curve of Suzuki type

In this paper, Algebraic-Geometric (AG) codes and quantum codes associated to a family of curves which comprises the famous Suzuki curve are investigated. The Weierstrass semigroup at some rational point is computed. Notably, each curve in the family turn out to be a Castle curve over some finite field, and a weak Castle curve over its extensions. This is a relevant feature when codes constructed from the curve are considered.

math.CO↗

arXiv2022

Minimal codewords in Norm-Trace codes

In this paper, we consider the affine variety codes obtained evaluating the polynomials $by=a_kx^k+\dots+a_1x+a_0$, $b,a_i\in\mathbb{F}_{q^r}$, at the affine $\F_{q^r}$-rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolutely irreducibility of certain algebraic varieties.

math.AG↗

arXiv2021

A generalization of Bring's curve in any characteristic

Let $p\ge 7$ be a prime, and $m\ge 5$ an integer. A natural generalization of Bring's curve valid over any field $\mathbb{K}$ of zero characteristic or positive characteristic $p$, is the algebraic variety $V$ of $\textrm{PG}(m-1,\mathbb{K})$ which is the complete intersection of the projective algebraic hypersurfaces of homogeneous equations $x_1^k+\cdots +x_m^{k}=0$ with $1\leq k\leq m-2$. In positive characteristic, we also assume $m\le p-1$. Up to a change of coordinates in $\textrm{PG}(m-1,\mathbb{K})$, we show that $V$ is a projective, absolutely irreducible, non-singular curve of $\textrm{PG}(m-2,\mathbb{K})$ with degree $(m-2)!$, genus $\mathfrak{g}= \frac{1}{4} ((m-2)(m-3)-4)(m-2)!+1$, and tame automorphism group $G$ isomorphic to $\textrm{Sym}_m$. We compute the genera of the quotient curves of $V$ with respect to the stabilizers of one or more coordinates under the action of $G$. In positive characteristic, the two extremal cases, $m=5$ and $m=p-1$ are investigated further. For $m=5$, we show that there exist infinitely many primes $p$ such that $V$ is $\mathbb{F}_{p^2}$-maximal curve of genus $4$. The smallest such primes are $29,59,149,239,839$. For $m=p-1$ we prove that $V$ has as many as $(p-2)!$ points over $\mathbb{F}_p$ and has no further points over $\mathbb{F}_{p^2}$. We also point out a connection with previous work of R\'edei about the famous Minkowski conjecture proven by Haj\'os (1941), as well as with a more recent result of Rodr\'iguez Villegas, Voloch and Zagier (2001) on plane curves attaining the St\"ohr-Voloch bound, and the regular sequence problem for systems of diagonal equations introduced by Conca, Krattenthaler and Watanabe (2009).

math.AG↗

arXiv2021

On a family of linear MRD codes with parameters $[8\times8,16,7]_q$

In this paper we consider a family $\mathcal{F}$ of $16$-dimensional $\mathbb{F}_q$-linear rank metric codes in $\mathbb{F}_q^{8\times8}$, arising from the polynomial $x^{q^s}+\delta x^{q^{4+s}}\in\mathbb{F}_{q^8}[x]$. Examples of MRD codes in $\mathcal{F}$ have been provided by Csajb\'ok, Marino, Polverino and Zanella (2018). For any large enough odd $q$, we determine exactly which codes in $\mathcal{F}$ are MRD. We also show that the MRD codes in $\mathcal{F}$ are not equivalent to any other MRD codes known so far.

math.CO↗

arXiv2021

PIR codes from combinatorial structures

A $k$-server Private Information Retrieval (PIR) code is a binary linear $[m,s]$-code admitting a generator matrix such that for every integer $i$ with $1\le i\le s$ there exist $k$ disjoint subsets of columns (called recovery sets) that add up to the vector of weight one, with the single $1$ in position $i$. As shown in \cite{Fazeli1}, a $k$-server PIR code is useful to reduce the storage overhead of a traditional $k$-server PIR protocol. Finding $k$-server PIR codes with a small blocklength for a given dimension has recently become an important research challenge. In this work, we propose new constructions of PIR codes from combinatorial structures, introducing the notion of $k$-partial packing. Several bounds over the existing literature are improved.

cs.IT↗

arXiv2021

On a conjecture on APN permutations

The single trivariate representation proposed in [C. Beierle, C. Carlet, G. Leander, L. Perrin, A Further Study of Quadratic APN Permutations in Dimension Nine, arXiv:2104.08008] of the two sporadic quadratic APN permutations in dimension 9 found by Beierle and Leander \cite{Beierle} is further investigated. In particular, using tools from algebraic geometry over finite fields, we prove that such a family does not contain any other APN permutation for larger dimensions.

cs.IT↗

arXiv2020

On the weight distribution of some minimal codes

Minimal codes are a class of linear codes which gained interest in the last years, thanks to their connections to secret sharing schemes. In this paper we provide the weight distribution and the parameters of families of minimal codes recently introduced by C. Tang, Y. Qiu, Q. Liao, Z. Zhou, answering some open questions.

math.CO↗