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GIOVANNI ZINI

Ricercatore t.d. art. 24 c. 3 lett. B
Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

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Pubblicazioni

2023 - Linear Maximum Rank Distance Codes of Exceptional Type [Articolo su rivista]
Bartoli, D.; Zini, G.; Zullo, F.
abstract

Scattered polynomials of a given index over finite fields are intriguing rare objects with many connections within mathematics. Of particular interest are the exceptional ones, as defined in 2018 by the first author and Zhou, for which partial classification results are known. In this paper we propose a unified algebraic description of F(q)n-linear maximum rank distance codes, introducing the notion of exceptional linear maximum rank distance codes of a given index. Such a connection naturally extends the notion of exceptionality for a scattered polynomial in the rank metric framework and provides a generalization of Moore sets in the monomial MRD context. We move towards the classification of exceptional linear MRD codes, by showing that the ones of index zero are generalized Gabidulin codes and proving that in the positive index case the code contains an exceptional scattered polynomial of the same index.

2023 - On a family of linear MRD codes with parameters [8 × 8 , 16 , 7] q [Articolo su rivista]
Timpanella, M.; Zini, G.
abstract

In this paper we consider a family F of 2n-dimensional F-q-linear rank metric codes in F-q(nxn) arising from polynomials of the form x(qs) +delta x(q) (n/2 +s) is an element of F-q(n) [x]. The family F was introduced by Csajbok et al. (JAMA 548:203-220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8, and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8, providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F-q-rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far.

2022 - Investigating the exceptionality of scattered polynomials [Articolo su rivista]
Bartoli, D.; Zini, G.; Zullo, F.
abstract

Scattered polynomials over a finite field Fqjavax.xml.bind.JAXBElement@46419929 have been introduced by Sheekey in 2016, and a central open problem regards the classification of those that are exceptional. So far, only two families of exceptional scattered polynomials are known. Very recently, Longobardi and Zanella weakened the property of being scattered by introducing the notion of L-qt-partially scattered and R-qt-partially scattered polynomials, for t a divisor of n. Indeed, a polynomial is scattered if and only if it is both L-qt-partially scattered and R-qt-partially scattered. In this paper, by using techniques from algebraic geometry over finite fields and function fields theory, we show that the property which is the hardest to be preserved is the L-qt-partially scattered one. We investigate a large family F of R-qt-partially scattered polynomials, containing examples of exceptional R-qt-partially scattered polynomials, which turn out to be connected with linear sets of so-called pseudoregulus type. We introduce two different notions of equivalence preserving the property of being R-qt-partially scattered. Many polynomials in F are inequivalent and geometric arguments are used to determine their equivalence classes under the action of ΓL(2n/t,qt).

2022 - Non-minimum tensor rank Gabidulin codes [Articolo su rivista]
Bartoli, D.; Zini, G.; Zullo, F.
abstract

The tensor rank of some Gabidulin codes of small dimension is investigated. In particular, we determine the tensor rank of any rank metric code equivalent to an 8-dimensional Fq-linear generalized Gabidulin code in Fq4×4. This shows that such a code is never minimum tensor rank. In this way, we detect the first infinite family of Gabidulin codes which are not minimum tensor rank.

2022 - On ideals in group algebras: An uncertainty principle and the Schur product [Articolo su rivista]
Borello, M.; Willems, W.; Zini, G.
abstract

In this paper, we investigate some properties of ideals in group algebras of finite groups over fields. First, we highlight an important link between their dimension, their minimal Hamming distance and the group order. This is a generalized version of an uncertainty principle shown in 1992 by Meshulam. Secondly, we introduce the notion of the Schur product of ideals in group algebras and investigate the module structure and the dimension of the Schur square. We give a structural result on ideals that coincide with their Schur square, and we provide conditions for an ideal to be such that its Schur square has the projective cover of the trivial module as a direct summand. This has particularly interesting consequences for group algebras of p-groups over fields of characteristic p.

2022 - On monomial generalized almost perfect nonlinear functions [Articolo su rivista]
Bartoli, D.; Giulietti, M.; Peraro, G.; Zini, G.
abstract

Generalized almost perfect nonlinear (GAPN) functions are a generalization of APN functions to finite fields of odd characteristic p introduced in 2017 by Kuroda and Tsujie. In this paper we deal with GAPN functions of monomial type. To this aim, we connect the GAPN property for a monomial function over Fpjavax.xml.bind.JAXBElement@7f2e9ed8 to the existence of suitable rational points of an algebraic curve defined over Fpjavax.xml.bind.JAXBElement@425705. We give necessary conditions for a monomial function to be GAPN, providing the converse of recent results by Özbudak and Sălăgean and by Zha, Hu and Zhang.

2022 - r-fat linearized polynomials over finite fields [Articolo su rivista]
Bartoli, D.; Micheli, G.; Zini, G.; Zullo, F.
abstract

r-fat polynomials are a natural generalization of scattered polynomials. They define linear sets of the projective line PG(1,qn) of rank n with r points of weight larger than one. Using techniques on algebraic curves and function fields, we obtain numerical bounds for r and the non-existence of exceptional r-fat polynomials with r>0. We completely determine the possible values of r when considering linearized polynomials over Fqjavax.xml.bind.JAXBElement@3a225500 and we also provide one family of 1-fat polynomials in PG(1,q5). Furthermore, we investigate LP-polynomials (i.e. polynomials of type f(x)=x+δxqjavax.xml.bind.JAXBElement@11206662∈Fqjavax.xml.bind.JAXBElement@16c0ec2a[x], gcd⁡(n,s)=1), determining the spectrum of values r for which such polynomials are r-fat.

2021 - COPRIME COMMUTATORS in the SUZUKI GROUPS [Articolo su rivista]
Zini, G.
abstract

In this note we show that every element of a simple Suzuki group is a commutator of elements of coprime orders.

2021 - Linear sets from projection of Desarguesian spreads [Articolo su rivista]
Napolitano, V.; Polverino, O.; Zini, G.; Zullo, F.
abstract

Every linear set in a projective space is the projection of a subgeometry, and most known characterizations of linear sets are given under this point of view. For instance, scattered linear sets of pseudoregulus type are obtained by considering a Desarguesian spread of a subgeometry and projecting from a vertex which is spanned by all but two director spaces. In this paper we introduce the concept of linear sets of h-pseudoregulus type, which turns out to be projected from the span of an arbitrary number of director spaces of a Desarguesian spread of a subgeometry. Among these linear sets, we characterize those which are h-scattered and solve the equivalence problem between them; a key role is played by an algebraic tool recently introduced in the literature and known as Moore exponent set. As a byproduct, we classify asymptotically h-scattered linear sets of h-pseudoregulus type.

2021 - On certain linearized polynomials with high degree and kernel of small dimension [Articolo su rivista]
Polverino, O.; Zini, G.; Zullo, F.
abstract

Let f be the Fq-linear map over Fqjavax.xml.bind.JAXBElement@3e829f17 defined by x↦x+axqjavax.xml.bind.JAXBElement@df90dc4+bxqjavax.xml.bind.JAXBElement@5c033a86 with gcd⁡(n,s)=1. It is known that the kernel of f has dimension at most 2, as proved by Csajbók et al. in [9]. For n big enough, e.g. n≥5 when s=1, we classify the values of b/a such that the kernel of f has dimension at most 1. To this aim, we translate the problem into the study of some algebraic curves of small degree with respect to the degree of f; this allows to use intersection theory and function field theory together with the Hasse-Weil bound. Our result implies a non-scatteredness result for certain high degree scattered binomials, and the asymptotic classification of a family of rank metric codes.

2021 - On certain self-orthogonal AG codes with applications to Quantum error-correcting codes [Articolo su rivista]
Bartoli, D.; Montanucci, M.; Zini, G.
abstract

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. Examples are given by Castle curves, GK curves, generalized GK curves and the Abdón–Bezerra–Quoos maximal curves. Applications of our method to these curves are provided. Our construction extends a previous one due to Hernando, McGuire, Monserrat, and Moyano-Fernández.

2021 - On the intersection problem for linear sets in the projective line [Articolo su rivista]
Zini, G.; Zullo, F.
abstract

The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with possibly different maximum fields of linearity. We also consider the intersection between a certain linear set of maximum rank and any other linear set of the same rank. The strategy relies on the study of certain algebraic curves whose rational points describe the intersection of the two linear sets. Among other geometric and algebraic tools, function field theory and the Hasse–Weil bound play a crucial role. As an application, we give asymptotic results on semifields of BEL-rank two.

2021 - On two Möbius functions for a finite non-solvable group [Articolo su rivista]
Dalla Volta, F.; Zini, G.
abstract

Let G be a finite group, μ be the Möbius function on the subgroup lattice of G, and λ be the Möbius function on the poset of conjugacy classes of subgroups of G. It was proved by Pahlings that, whenever G is solvable, the property (Formula presented.) holds for any subgroup H of G. It is known that this property does not hold in general, the Mathieu group M 12 being a counterexample. In this paper we investigate the relation between μ and λ for some classes of non-solvable groups, among them, the minimal non-solvable groups. We also provide several examples of groups not satisfying the property.

2021 - Scattered subspaces and related codes [Articolo su rivista]
Zini, G.; Zullo, F.
abstract

After a seminal paper by Shekeey (Adv Math Commun 10(3):475-488, 2016), a connection between maximum h-scattered Fq-subspaces of V(r, qn) and maximum rank distance (MRD) codes has been established in the extremal cases h= 1 and h= r- 1. In this paper, we propose a connection for any h∈ { 1 , … , r- 1 } , extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. We show that, up to equivalence, MRD codes having the same parameters as the ones in our connection come from an h-scattered subspace. Also, we determine the weight distribution of codes related to the geometric counterpart of maximum h-scattered subspaces.

2021 - The Mobius function of PSL (3, 2 p) for any prime p [Articolo su rivista]
Borello, M.; Dalla Volta, F.; Zini, G.
abstract

Let G be the simple group PSL(3, 2p), where p is a prime number. For any subgroup H of G, we compute the Mobius function μ(H) of H in the subgroup lattice of G. To this aim, we describe the intersections of maximal subgroups of G. We point out some connections of the Mobius function with other combinatorial objects, and, in this context, we compute the reduced Euler characteristic of the order complex of the subposet of r-subgroups of PGL(3,q), for any prime r and any prime power q.

2021 - Weierstrass semigroups at every point of the Suzuki curve [Articolo su rivista]
Bartoli, D.; Montanucci, M.; Zini, G.
abstract

2020 - On plane curves given by separated polynomials and their automorphisms [Articolo su rivista]
Bonini, M.; Montanucci, M.; Zini, G.
abstract

Let ?"' be a plane curve defined over the algebraic closure K of a finite prime field ?"½p by a separated polynomial, that is ?"' : A(Y) = B(X), where A(Y) is an additive polynomial of degree pn and the degree m of B(X) is coprime with p. Plane curves given by separated polynomials are widely studied; however, their automorphism groups are not completely determined. In this paper we compute the full automorphism group of ?"' when m a‰¢ 1 mod pn and B(X) = Xm. Moreover, some sufficient conditions for the automorphism group of ?"' to imply that B(X) = Xm are provided. Also, the full automorphism group of the norm-trace curve ?"' : X(qjavax.xml.bind.JAXBElement@42c3fbd6-1)/(q-1) = Yqjavax.xml.bind.JAXBElement@2638b9c6 + Yqjavax.xml.bind.JAXBElement@3cec48ce + ... + Y is computed. Finally, these results are used to show that certain one-point AG codes have many automorphisms.

2020 - Quotients of the Hermitian curve from subgroups of PGU (3 , q) without fixed points or triangles [Articolo su rivista]
Montanucci, M.; Zini, G.
abstract

In this paper, we deal with the problem of classifying the genera of quotient curves Hq/ G, where Hq is the Fq2-maximal Hermitian curve and G is an automorphism group of Hq. The groups G considered in the literature fix either a point or a triangle in the plane PG (2 , q6). In this paper, we give a complete list of genera of quotients Hq/ G, when G≤ Aut (Hq) ≅ PGU (3 , q) does not leave invariant any point or triangle in the plane. Also, the classification of subgroups G of PGU (3 , q) satisfying this property is given up to isomorphism.

2020 - The complete list of genera of quotients of the Fqjavax.xml.bind.JAXBElement@65d340c9-maximal Hermitian curve for q ≡ 1 (mod 4) [Articolo su rivista]
Montanucci, M.; Zini, G.
abstract

Let Fqjavax.xml.bind.JAXBElement@60d37c4e be the finite field with q2 elements. Most of the known Fqjavax.xml.bind.JAXBElement@467809f9-maximal curves arise as quotient curves of the Fqjavax.xml.bind.JAXBElement@2829bbbf-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 completely the spectrum of genera of quotients of Hq for any q≡1(mod4).

2019 - On the classification problem for the genera of quotients of the Hermitian curve [Articolo su rivista]
Dalla Volta, F.; Montanucci, M.; Zini, G.
abstract

In this article, we characterize the genera of those quotient curves Hq/G of the Fqjavax.xml.bind.JAXBElement@45f83521-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 associated to Hq(Fqjavax.xml.bind.JAXBElement@590af847) In this way, several new values for the genus of a maximal curve over a finite field are obtained. Our results leave just two open cases to provide the complete list of genera of Galois subcovers of the Hermitian curve; namely, the open cases in [4] when G fixes a point P ϵ Hq(Fqjavax.xml.bind.JAXBElement@5b677e2a) and q is even, and the open cases in [33] when G≤M2 and q is odd.

2019 - The Möbius function of PSU(3, 22n) [Articolo su rivista]
Zini, G.
abstract

Let G be the simple group PSU(3, 22n), n > 0. For any subgroup H of G, we compute the Möbius function µL(H, G) of H in the subgroup lattice L of G, and the Möbius function µL¯ ([H], [G]) of [H] in the poset L¯ of conjugacy classes of subgroups of G. For any prime p, we provide the Euler characteristic of the order complex of the poset of non-trivial p-subgroups of G.

2018 - AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves [Articolo su rivista]
Montanucci, M.; Timpanella, M.; Zini, G.
abstract

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-rational point is shown to be symmetric.

2018 - AG codes and AG quantum codes from the GGS curve [Articolo su rivista]
Bartoli, D.; Montanucci, M.; Zini, G.
abstract

In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all Fq2-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 with better relative parameters with respect to one-point Hermitian codes are discovered. Classes of quantum and convolutional codes are provided relying on the constructed AG codes.

2018 - Algebraic geometric codes on many points from Kummer extensions [Articolo su rivista]
Bartoli, D.; Quoos, L.; Zini, G.
abstract

For Kummer extensions defined by ym=f(x), where f(x) is a separable polynomial over the finite field Fq, we compute the number of Weierstrass gaps at two totally ramified places. For many totally ramified places we give a criterion to find pure gaps at these points and present families of pure gaps. We then apply our results to construct n-points algebraic geometric codes with good parameters.

2018 - Multi point AG codes on the GK maximal curve [Articolo su rivista]
Bartoli, D.; Montanucci, M.; Zini, G.
abstract

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.

2018 - On permutation trinomials of type x2pjavax.xml.bind.JAXBElement@63a01e08+r+xpjavax.xml.bind.JAXBElement@41e90317+r+λxr [Articolo su rivista]
Bartoli, D.; Zini, G.
abstract

We determine all permutation trinomials of type x2pjavax.xml.bind.JAXBElement@2e2c1362+r+xpjavax.xml.bind.JAXBElement@285c7208+r+λxr over the finite field Fpjavax.xml.bind.JAXBElement@4bef7955 when (2ps+r)4

2018 - On some Galois covers of the Suzuki and Ree curves [Articolo su rivista]
Giulietti, M.; Montanucci, M.; Quoos, L.; Zini, G.
abstract

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 Fqjavax.xml.bind.JAXBElement@55548774, and R˜q is not Galois covered by the Hermitian curve maximal over Fqjavax.xml.bind.JAXBElement@4d8da0d9. We also compute the genera of many Galois subcovers of S˜q and R˜q; in this way, many new values in the spectrum of genera of maximal curves are obtained. The full automorphism group of both S˜q and R˜q is determined.

2018 - On the spectrum of genera of quotients of the Hermitian curve [Articolo su rivista]
Montanucci, M.; Zini, G.
abstract

We investigate the genera of quotient curves ℋq∕G of the Fq2 -maximal Hermitian curve ℋq, where G is contained in the maximal subgroup Mq ≤ Aut (Hq) fixing a pole-polar pair (P,ℓ) with respect to the unitary polarity associated with ℋq. To this aim, a geometric and group-theoretical description of ℳq is given. The genera of some other quotients ℋq∕G with G≰ℳq are also computed. In this way we obtain new values in the spectrum of genera of Fq2 -maximal curves. The complete list of genera g>1 of quotients of ℋq is given for q≤29, as well as the genera g of quotients of ℋq with g>q2q+30/32 for any q. As a direct application, we exhibit examples of Fq2 -maximal curves which are not Galois covered by ℋq when q is not a cube. Finally, a plane model for ℋq∕G is obtained when G is cyclic of order p⋅d, with d a divisor of q+1.

2017 - Complete (k,4) -arcs from quintic curves [Articolo su rivista]
Bartoli, D.; Speziali, P.; Zini, G.
abstract

Complete (k, 4)-arcs in projective Galois planes are the geometric counterpart of linear non-extendible codes of length k, dimension 3 and Singleton defect 2. A class of infinite families of complete (k, 4)-arcs in PG (2 , q) is constructed, for q a power of an odd prime p≡3(mod4), p> 3. The order of magnitude of k is smaller than q. This property significantly distinguishes the complete (k, 4)-arcs of this paper from the previously known infinite families, whose size exceeds q-6q.

2017 - Complete permutation polynomials from exceptional polynomials [Articolo su rivista]
Bartoli, D.; Giulietti, M.; Quoos, L.; Zini, G.
abstract

We classify complete permutation monomials of degree qn−1q−1+1 over the finite field with qn elements in odd characteristic, for n+1 a prime and (n+1)4

2017 - Generalized Artin–Mumford curves over finite fields [Articolo su rivista]
Montanucci, M.; Zini, G.
abstract

Let Fq be the finite field of order q=ph with p>2 prime and h>1, and let Fq¯ be a subfield of Fq. From any two q¯-linearized polynomials L1,L2∈F‾q[T] of degree q, we construct an ordinary curve X(Ljavax.xml.bind.JAXBElement@3c782d3,Ljavax.xml.bind.JAXBElement@42bc9a9f) of genus g=(q−1)2 which is a generalized Artin–Schreier cover of the projective line P1. The automorphism group of X(Ljavax.xml.bind.JAXBElement@7943b174,Ljavax.xml.bind.JAXBElement@3b2e0dbf) over the algebraic closure F‾q of Fq contains a semidirect product Σ⋊Γ of an elementary abelian p-group Σ of order q2 by a cyclic group Γ of order q¯−1. We show that for L1≠L2, Σ⋊Γ is the full automorphism group Aut(X(Ljavax.xml.bind.JAXBElement@4c0d113,Ljavax.xml.bind.JAXBElement@228ecb76)) over F‾q; for L1=L2 there exists an extra involution and Aut(X(Ljavax.xml.bind.JAXBElement@46220642,Ljavax.xml.bind.JAXBElement@daaa8cc))=Σ⋊Δ with a dihedral group Δ of order 2(q¯−1) containing Γ. Two different choices of the pair {L1,L2} may produce birationally isomorphic curves, even for L1=L2. We prove that any curve of genus (q−1)2 whose F‾q-automorphism group contains an elementary abelian subgroup of order q2 is birationally equivalent to X(Ljavax.xml.bind.JAXBElement@f60a823,Ljavax.xml.bind.JAXBElement@5b6117c2) for some separable q¯-linearized polynomials L1,L2 of degree q. We produce an analogous characterization in the special case L1=L2. This extends a result on the Artin–Mumford curves, due to Arakelian and Korchmáros [1].

2017 - Some Ree and Suzuki curves are not Galois covered by the Hermitian curve [Articolo su rivista]
Montanucci, M.; Zini, G.
abstract

The Deligne–Lusztig curves associated to the algebraic groups of type A22, B22, and G22 are classical examples of maximal curves over finite fields. The Hermitian curve Hq is maximal over Fqjavax.xml.bind.JAXBElement@30720d30, for any prime power q, the Suzuki curve Sq is maximal over Fqjavax.xml.bind.JAXBElement@79adec00, for q=22h+1, h≥1, and the Ree curve Rq is maximal over Fqjavax.xml.bind.JAXBElement@5add848c, for q=32h+1, h≥0. In this paper we show that S8 is not Galois covered by H64. We also prove an unpublished result due to Rains and Zieve stating that R3 is not Galois covered by H27. Furthermore, we determine the spectrum of genera of Galois subcovers of H27, and we point out that some Galois subcovers of R3 are not Galois subcovers of H27.

2016 - Complete (k,3)-arcs from quartic curves [Articolo su rivista]
Bartoli, D.; Giulietti, M.; Zini, G.
abstract

Complete (Formula presented.) -arcs in projective planes over finite fields are the geometric counterpart of linear non-extendible Near MDS codes of length (Formula presented.) and dimension (Formula presented.). A class of infinite families of complete (Formula presented.) -arcs in (Formula presented.) is constructed, for (Formula presented.) a power of an odd prime (Formula presented.). The order of magnitude of (Formula presented.) is smaller than (Formula presented.). This property significantly distinguishes the complete (Formula presented.) -arcs of this paper from the previously known infinite families, whose size differs from (Formula presented.) by at most (Formula presented.).

2016 - Maximal curves from subcovers of the GK-curve [Articolo su rivista]
Giulietti, M.; Quoos, L.; Zini, G.
abstract

For every q=n3 with n a prime power greater than 2, the GK-curve is an Fqjavax.xml.bind.JAXBElement@65481bd0-maximal curve that is not Fqjavax.xml.bind.JAXBElement@5a5eaa49-covered by the Hermitian curve. In this paper some Galois subcovers of the GK curve are investigated. Infinitely many examples of maximal curves that cannot be Galois covered by the Hermitian curve are obtained. We also describe explicit equations for some families of quotient curves of the GK-curve. In several cases, such curves provide new values in the spectrum of genera of Fqjavax.xml.bind.JAXBElement@4cf3c39c-maximal curves.

2016 - On maximal curves that are not quotients of the Hermitian curve [Articolo su rivista]
Giulietti, M.; Montanucci, M.; Zini, G.
abstract

For each prime power ℓ the plane curve Xℓ with equation Yℓ2ℓ+1=Xℓ2-X is maximal over Fℓ6. Garcia and Stichtenoth in 2006 proved that X3 is not Galois covered by the Hermitian curve and raised the same question for Xℓ with ℓ>3; in this paper we show that Xℓ is not Galois covered by the Hermitian curve for any ℓ>3. Analogously, Duursma and Mak proved that the generalized GK curve Cℓn over Fℓ2n is not a quotient of the Hermitian curve for ℓ>2 and n≥5, leaving the case ℓ=2 open; here we show that C2n is not Galois covered by the Hermitian curve over F22n for n≥5.

2016 - On monomial complete permutation polynomials [Articolo su rivista]
Bartoli, D.; Giulietti, M.; Zini, G.
abstract

We investigate monomials axd over the finite field with q elements Fq, in the case where the degree d is equal to +1 with q=(q′)n for some n. For n=6 we explicitly list all a's for which axd is a complete permutation polynomial (CPP) over Fq. Some previous characterization results by Wu et al. for n=4 are also made more explicit by providing a complete list of a's such that axd is a CPP. For odd n, we show that if q is large enough with respect to n then axd cannot be a CPP over Fq, unless q is even, n≡3(mod4), and the trace TrFjavax.xml.bind.JAXBElement@4b875630/Fjavax.xml.bind.JAXBElement@12e7e698(a−1) is equal to 0.