Maria Montanucci

Maria Montanucci
Technical University of Denmark | DTU · Department of Applied Mathematics and Computer Science

Ph.D. (“Doctor Europaeus”) in Mathematics and Computer Science

About

61
Publications
2,275
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290
Citations
Additional affiliations
November 2018 - February 2019
University of Padova
Position
  • PostDoc Position
November 2015 - November 2018
Università degli Studi della Basilicata
Position
  • PhD Student
Education
September 2013 - July 2015
Università degli Studi di Perugia
Field of study
  • Mathematics
September 2010 - July 2013
Università degli Studi di Perugia
Field of study
  • Mathematics

Publications

Publications (61)
Article
We prove that under the action of GL(2,q6) there are ⌊(q2+q+1)(q−2)/2⌋ equivalence classes of maximum scattered subspaces of the form Ub={(x,bxq+xq4):x∈Fq6} in Fq6×Fq6. This verifies a conjecture of Csajbók, Marino, Polverino and Zanella from 2018.
Article
Maximum scattered linear sets in PG(1,qn) have been completely classified for n≤4, see Csajbók and Zanella (2018) [9]; Lavrauw and Van de Voorde (2010) [10]. Here a wide class of linear sets in PG(1,q5) is studied which depends on two parameters. Conditions for the existence, in this class, of possible new maximum scattered linear sets in PG(1,q5)...
Preprint
Full-text available
In this paper we determine the generalized Weierstrass semigroup $ \widehat{H}(P_{\infty}, P_1, \ldots , P_{m})$, and consequently the Weierstrass semigroup $H(P_{\infty}, P_1, \ldots , P_{m})$, at $m+1$ points on the curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$. These curves has been introduced in Tafazolian et al. as new examples of cur...
Preprint
Let $\mathcal{X}$ be a projective, irreducible, nonsingular algebraic curve over the finite field $\mathbb{F}_q$ with $q$ elements and let $|\mathcal{X}(\mathbb{F}_q)|$ and $g(\mathcal X)$ be its number of rational points and genus respectively. The Ihara constant $A(q)$ has been intensively studied during the last decades, and it is defined as the...
Article
Full-text available
Let X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}$$\end{document} be an irreducible, non-singular, algebraic curve defined over a field of odd charact...
Article
In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X ⁶ + Y ⁶ + ℨ ⁶ +( X ² + Y ² + ℨ ² )( X ⁴ + Y ⁴ + ℨ ⁴ )−12 X ² Y ² ℨ ² = 0, and showed that its full automorphism group is isomorphic to the symmetric group S 5 . We show that this holds also over every algebraically closed field 𝕂 of characte...
Article
In [14], D. Skabelund constructed a maximal curve over Fq4 as a cyclic cover of the Suzuki curve. In this paper we explicitly determine the structure of the Weierstrass semigroup at any point P of the Skabelund curve. We show that its Weierstrass points are precisely the Fq4-rational points. Also we show that among the Weierstrass points, two types...
Article
Full-text available
In this paper a construction of quantum codes from self-orthogonal algebraic geometry codes is provided. Our method is based on the CSS construction as well as on some peculiar properties of the underlying algebraic curves, named Swiss curves. Several classes of well-known algebraic curves with many rational points turn out to be Swiss curves. Exam...
Preprint
In 2017 Skabelund constructed two new examples of maximal curves $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ as covers of the Suzuki and Ree curves, respectively. The resulting Skabelund curves are analogous to the Giulietti-Korchm\'aros cover of the Hermitian curve. In this paper a complete characterization of all Galois subcovers of the S...
Article
Let f(X)∈Fqr[X] be a q-polynomial. If the Fq-subspace U={(xqt,f(x))|x∈Fqn} defines a maximum scattered linear set, then we call f(X) a scattered polynomial of index t. The asymptotic behavior 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 max...
Article
In this paper, explicit equations for algebraic curves with genus 4, 5, and 10 already studied in characteristic zero, are analyzed in positive characteristic p. We show that these curves have an interesting behaviour on the number of their rational places. Namely, they are either maximal or minimal over the finite field with p2 elements for infini...
Article
Full-text available
In this paper we deal with the problem of classifying the genera of quotient curves $\mathcal{H}_q/G$, where $\mathcal{H}_q$ is the $\mathbb{F}_{q^2}$-maximal Hermitian curve and $G$ is an automorphism group of $\mathcal{H}_q$. The groups $G$ considered in the literature fix either a point or a triangle in the plane ${\rm PG}(2,q^6)$. In this paper...
Preprint
For a non-degenerate irreducible curve $C$ of degree $d$ in $\mathbb{P}^3$ over $\mathbb{F}_q$, we prove that the number $N_q(C)$ of $\mathbb{F}_q$-rational points of $C$ satisfies the inequality $N_q(C) \leq (d-2)q+1$. Our result improves the previous bound $N_q(C) \leq (d-1)q+1$ obtained by Homma in 2012 and leads to a natural conjecture generali...
Article
Full-text available
The second generalized GK function fields Kn are a recently found family of maximal function fields over the finite field with q2n elements, where q is a prime power and n≥1 an odd integer. In this paper we construct many new maximal function fields by determining various Galois subfields of Kn. In case gcd⁡(q+1,n)=1 and either q is even or q≡1(mod...
Article
A Locally Recoverable Code is a code such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. When we have δ non-overlapping subsets of cardinality ri that can be used to recover the missing coordinate we say that a linear code C with length n, dimension k, minimum distance...
Article
Let q be a prime-power, and n≥3 an odd integer. We determine the structure of the Weierstrass semigroup H(P) where P is an arbitrary Fq2-rational point of GK2,n where GK2,n stands for the Fq2n-maximal curve of Beelen and Montanucci. We prove that these points are Weierstrass points, and we compute the Frobenius dimension of GK2,n. Using these resul...
Article
Let Fq2 be the finite field with q2 elements. Most of the known Fq2-maximal curves arise as quotient curves of the Fq2-maximal Hermitian curve Hq. After a seminal paper by Garcia, Stichtenoth and Xing, many papers have provided genera of quotients of Hq, but their complete determination is a challenging open problem. In this paper we determine comp...
Preprint
Full-text available
In 2017, D. Skabelund constructed a maximal curve over $\mathbb{F}_{q^4}$ as a cyclic cover of the Suzuki curve. In this paper we explicitly determine the structure of the Weierstrass semigroup at any point $P$ of the Skabelund curve. We show that its Weierstrass points are precisely the $\mathbb{F}_{q^4}$-rational points. Also we show that among t...
Preprint
Maximum scattered subspaces are not only objects of intrinsic interest in finite geometry but also powerful tools for the construction of MRD-codes, projective two-weight codes, and strongly regular graphs. In 2018 Csajb\'ok, Marino, Polverino, and Zanella introduced a new family of maximum scattered subspaces in $\mathbb{F}_{q^6} \times \mathbb{F}...
Article
In [10], the existence of Fq-linear MRD-codes of Fq6×6, with dimension 12, minimum distance 5 and left idealiser isomorphic to Fq6, defined by a trinomial of Fq6[x], when q is odd and q≡0,±1(mod5), has been proved. In this paper we show that this family produces Fq-linear MRD-codes of Fq6×6, with the same properties, also in the remaining q odd cas...
Article
Let X be a (projective, nonsingular, geometrically irreducible) curve of even genus g(X)≥2 defined over an algebraically closed field K of odd characteristic p. If the p-rank γ(X) equals g(X), then X is ordinary. In this paper, we deal with large automorphism groups G of ordinary curves of even genus. We prove that |G|<821.37g(X)7/4. The proof of o...
Preprint
Let $\mathcal{X}$ be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic $p$. Let $g$ and $\gamma$ be the genus and $p$-rank of $\mathcal{X}$, respectively. The influence of $g$ and $\gamma$ on the automorphism group $Aut(\mathcal{X})$ of $\mathcal{X}$ is well-known in the literature. If $g \geq 2$ then $Aut(\ma...
Preprint
Full-text available
A Locally Recoverable Code is a code such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. When we have $\delta$ non overlapping subsets of cardinality $r_i$ that can be used to recover the missing coordinate we say that a linear code $\mathcal{C}$ with length $n$, dimens...
Preprint
In this paper a construction of quantum codes from self-orthogonal algebraic geometry codes is provided. Our method is based on the CSS construction as well as on some peculiar properties of the underlying algebraic curves, named Swiss curves. Several classes of well-known algebraic curves with many rational points turn out to be Swiss curves. Exam...
Article
Let X be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus g≥2 defined over an algebraically closed field K of odd characteristic p≥0, and let Aut(X) be the group of all automorphisms of X which fix K element-wise. For any a subgroup G of Aut(X) whose order is a power of an odd prime d other than p, the bound proven by...
Preprint
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 c...
Preprint
Full-text available
Let $\mathcal{X}$ be a (projective, non-singular, geometrically irreducible) curve of even genus $g(\mathcal{X}) \geq 2$ defined over an algebraically closed field $K$ of odd characteristic $p$. If the $p$-rank $\gamma(\mathcal{X})$ equals $g(\mathcal{X})$, then $\mathcal{X}$ is \emph{ordinary}. In this paper, we deal with \emph{large} automorphism...
Preprint
Full-text available
In [10], the existence of $\mathbb{F}_q$-linear MRD-codes of $\mathbb{F}_q^{6\times 6}$, with dimension $12$, minimum distance $5$ and left idealiser isomorphic to $\mathbb{F}_{q^6}$, defined by a trinomial of $\mathbb{F}_{q^6}[x]$, when $q$ is odd and $q\equiv 0,\pm 1\pmod 5$, has been proved. In this paper we show that this family produces $\math...
Preprint
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, excep...
Preprint
The maximum scattered linear sets in $PG(1,q^n)$ have been completely classified for $n \le 4$ by Csajb\'ok-Zanella and Lavrauw-Van de Voorde. Here a wide class of linear sets in $PG(1,q^5)$ is studied which depends on two parameters. Conditions for the existence, in this class, of possible new maximum scattered linear sets in $PG(1,q^5)$ are exhib...
Article
In this article, we characterize the genera of those quotient curves Hq/G of the Fq2-maximal Hermitian curve Hq for which either G is contained in the maximal subgroup M1 of Aut(Hq) fixing a self-polar triangle, or q is even and G is contained in the maximal subgroup M2 of Aut(Hq) fixing a pole-polar pair (P,ℓ) with respect to the unitary polarity...
Article
Full-text available
In this paper we construct functional codes from Denniston maximal arcs. For q=24ℓ+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=2^{4\ell +2}$$\end{document} we ob...
Preprint
Full-text available
The second generalized GK maximal curves $\mathcal{GK}_{2,n}$ are maximal curves over finite fields with $q^{2n}$ elements, where $q$ is a prime power and $n \geq 3$ an odd integer, constructed by Beelen and Montanucci. In this paper we determine the structure of the Weierstrass semigroup $H(P)$ where $P$ is an arbitrary $\mathbb{F}_{q^2}$-rational...
Preprint
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.
Preprint
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 mi...
Preprint
The second generalized GK function fields $K_n$ are a recently found family of maximal function fields over the finite field with $q^{2n}$ elements, where $q$ is a prime power and $n \ge 1$ an odd integer. In this paper we construct many new maximal function fields by determining various Galois subfields of $K_n$. In case $\gcd(q+1,n)=1$ and either...
Preprint
Full-text available
Let $\mathcal{X}$ be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus $g \ge 2$ defined over an algebraically closed field $\mathbb{K}$ of odd characteristic $p\ge 0$, and let $\rm{Aut}(\mathcal{X})$ be the group of all automorphisms of $\mathcal{X}$ which fix $\mathbb{K}$ element-wise. For any a subgroup $G$ of $\rm{...
Article
Full-text available
In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all $\mathbb F_{q^2}$-rational points of the curve is determined; the Feng-Rao designed minimum distance is computed for infinite families of such codes, as well as the automorphism group. As a result, some linear codes...
Preprint
Full-text available
Let $\mathbb{F}_{q^2}$ be the finite field with $q^2$ elements. Most of the known $\mathbb{F}_{q^2}$-maximal curves arise as quotient curves of the $\mathbb{F}_{q^2}$-maximal Hermitian curve $\mathcal{H}_{q}$. After a seminal paper by Garcia, Stichtenoth and Xing, many papers have provided genera of quotients of $\mathcal{H}_q$, but their complete...
Preprint
In this paper we characterize the genera of those quotient curves $\mathcal{H}_q/G$ of the $\mathbb{F}_{q^2}$-maximal Hermitian curve $\mathcal{H}_q$ for which $G$ is contained in the maximal subgroup $\mathcal{M}_1$ of ${\rm Aut}(\mathcal{H}_q)$ fixing a self-polar triangle, or $q$ is even and $G$ is contained in the maximal subgroup $\mathcal{M}_...
Preprint
In 1895 Wiman introduced a Riemann surface $\mathcal{W}$ of genus $6$ over the complex field $\mathbb{C}$ defined by the homogeneous equation $\mathcal{W}:X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2 Z^2=0$, and showed that its full automorphism group is isomorphic to the symmetric group $S_5$. The curve $\mathcal{W}$ was previously studied as...
Preprint
Full-text available
In 1895 Wiman introduced a Riemann surface $\mathcal{W}$ of genus $6$ over the complex field $\mathbb{C}$ defined by the homogeneous equation $\mathcal{W}:X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2 Z^2=0$, and showed that its full automorphism group is isomorphic to the symmetric group $S_5$. The curve $\mathcal{W}$ was previously studied a...
Article
Full-text available
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 ratio...
Article
Full-text available
We investigate several types of linear codes constructed from two families of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves. Plane models for such curves are provided, and the Weierstrass semigroup at an Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \use...
Article
We investigate two families S~q and R~q of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves. We show that S~q is not Galois covered by the Hermitian curve maximal over Fq4, and R~q is not Galois covered by the Hermitian curve maximal over Fq6. We also compute the genera of many Galois...
Article
Full-text available
In this paper we investigate multi-point Algebraic-Geometric codes associated to the GK maximal curve, starting from a divisor which is invariant under a large automorphism group of the curve. We construct families of codes with large automorphism groups.
Article
Full-text available
In this article we construct for any prime power $q$ and odd $n \ge 5$, a new $\mathbb{F}_{q^{2n}}$-maximal curve $\mathcal X_n$. Like the Garcia--G\" uneri--Stichtenoth maximal curves, our curves generalize the Giulietti--Korchm\'aros maximal curve, though in a different way. We compute the full automorphism group of $\mathcal X_n$, yielding that...
Article
Full-text available
In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Giulietti-Korchm\'aros curve $\mathcal{X}$. We show that as the point varies, exactly three possibilities arise: One for the $\mathbb{F}_{q^2}$-rational points (already known in the literature), one for the $\mathbb{F}_{q^6} \setminus...
Article
Let $\mathcal{C}$ be a plane curve defined over the algebraic closure $K$ of a prime finite field $\mathbb{F}_p$ by a separated polynomial, that is $\mathcal{C}: A(y)=B(x)$, where $A(y)$ is an additive polynomial of degree $p^n$ and the degree $m$ of $B(X)$ is coprime with $p$. Plane curves given by separated polynomials are well-known and studied...
Article
Full-text available
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-Korchm\'aros example shows provided that $q>...
Article
Full-text available
Let $\mathcal{X}$ be a (projective, non-singular, geometrically irreducible) curve defined over an algebraically closed field $K$ of characteristic $p$. If the $p$-rank $\gamma(\mathcal{X})$ equals the genus $g(\mathcal{X})$, then $\mathcal{X}$ is ordinary. In this paper, we deal with \emph{large} automorphism groups $G$ of ordinary curves. On the...
Article
Full-text available
We investigate the genera of quotient curves $\mathcal H_q/G$ of the $\mathbb F_{q^2}$-maximal Hermitian curve $\mathcal H_q$, where $G$ is contained in the maximal subgroup $\mathcal M_q\leq{\rm Aut}(\mathcal H_q)$ fixing a pole-polar pair $(P,\ell)$ with respect to the unitary polarity associated with $\mathcal H_q$. To this aim, a geometric and...
Article
Let $\mathbb{K}$ be the algebraic closure of a finite field $\mathbb{F}_q$ of odd characteristic $p$. For a positive integer $m$ prime to $p$, let $F=\mathbb{K}(x,y)$ be the transcendency degree $1$ function field defined by $y^q+y=x^m+x^{-m}$. Let $t=x^{m(q-1)}$ and $H=\mathbb{K}(t)$. The extension $F|H$ is a non-Galois extension. Let $K$ be the G...
Article
For a power $q$ of a prime $p$, the Artin-Schreier-Mumford curve $ASM(q)$ of genus $g=(q-1)^2$ is the nonsingular model $\mathcal{X}$ of the irreducible plane curve with affine equation $(X^q+X)(Y^q+Y)=c,\, c\neq 0,$ defined over a field $\mathbb{K}$ of characteristic $p$. The Artin-Schreier-Mumford curves are known from the study of algebraic curv...
Article
Let $\mathbb{F}_q$ be the finite field of order $q=p^h$ with $p>2$ prime and $h>1$, and let $\mathbb{F}_{\bar{q}}$ be a subfield of $\mathbb{F}_q$. From any two $\bar{q}$-linearized polynomials $L_1,L_2 \in \overline{\mathbb{F}}_q[T]$ of degree $q$, we construct an ordinary curve $\mathcal{X}_{(L_1,L_2)}$ of genus $g=(q-1)^2$ which is a generalized...
Article
Let $\mathcal{X}$ be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus $\mathcal{g}(\mathcal{X}) \ge 2$ defined over an algebraically closed field $\mathbb{K}$ of odd characteristic $p$. Let $Aut(\mathcal{X})$ be the group of all automorphisms of $\mathcal{X}$ which fix $\mathbb{K}$ element-wise. For any solvable subgr...
Article
Full-text available
We determine the full automorphism group of two recently constructed families $\tilde{\mathcal{S}}_q$ and $\tilde{\mathcal{R}}_q$ of maximal curves over finite fields. These curves are covers of the Suzuki and Ree curves, and are analogous to the Giulietti-Korchm\'aros cover of the Hermitian curve. We also show that $\tilde{\mathcal{S}}_q$ is not G...
Article
The Deligne-Lusztig curves associated to the algebraic groups of type $^2A_2$, $^2B_2$, and $^2G_2$ are classical examples of maximal curves over finite fields. The Hermitian curve $\mathcal H_q$ is maximal over $\mathbb F_{q^2}$, for any prime power $q$, the Suzuki curve $\mathcal S_q$ is maximal over $\mathbb F_{q^4}$, for $q=2^{2h+1}$, $h\geq1$...
Article
For an algebraic curve X defined over an algebraically closed field of characteristic p > 0, the a-number a(X) is the dimension of the space of exact holomorphic differentials on X. We compute the a-number for an infinite families of Fermat and Hurwitz curves. Our results apply to Hermitian curves giving a new proof for a previous result of Gross.
Article
For each prime power $\ell$ the plane curve $\mathcal X_\ell$ with equation $Y^{\ell^2-\ell+1}=X^{\ell^2}-X$ is maximal over $\mathbb{F}_{\ell^6}$. Garcia and Stichtenoth in 2006 proved that $\mathcal X_3$ is not Galois covered by the Hermitian curve and raised the same question for $\mathcal X_\ell$ with $\ell>3$; in this paper we show that $\math...

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Project (1)
Project
Obtaining new (and sharper) bounds for the size of the automorphism group of an ordinary curve in characteristic p >0.