Hermitian Curve Research Articles

On the automorphism group of a family of maximal curves not covered by the Hermitian curve

In this paper we compute the automorphism group of the curves Xa,b,n,s and Yn,s introduced in Tafazolian et al. [27] as new examples of maximal curves which cannot be covered by the Hermitian curve. They arise as subcovers of the first generalized GK curve (GGS curve). As a result, a new characterization of the GK curve, as a member of this family, is obtained.

Hamiltonian, geometric momentum and force operators for a spin zero particle on a curve: physical approach

The Hamiltonian for a spin zero particle that is confined to a 1D curve embedded in the 3D space is constructed. Confinement is achieved by starting with the particle living in a small tube surrounding the curve, and assuming an infinitely strong normal force that squeezes the thickness of the tube to zero, eventually pinning the particle to the curve. We follow the new approach that we applied to confine a particle to a surface, in that we start with an expression for the 3D momentum operators whose components along and normal to the curve directions are separately Hermitian. The kinetic energy operator expressed in terms of the momentum operator in the normal direction is then a Hermitian operator in this case. When this operator is dropped and the thickness of the tube surrounding the curve is set to zero, one automatically gets the Hermitian curve Hamiltonian that contains the geometric potential term as expected. It is demonstrated that the origin of this potential lies in the ordering or symmetrization of the original 3D momentum operators in order to render them Hermitian. The Hermitian momentum operator for the particle as it is confined to the curve is also constructed and is seen to be similar to what is known as the geometric momentum of a particle confined to a surface in that it has a term proportional to the curvature that is along the normal to the curve. The force operator of the particle on the curve is also derived, and is shown to reduce, for a curve with a constant curvature and torsion, to a -apparently- single component normal to the curve that is a symmetrization of the classical expression plus a quantum term. All the above quantities are then derived for the specific case of a particle confined to a cylindrical helix embedded in 3D space.

Galois subcovers of the Hermitian curve in characteristic p with respect to subgroups of order p2

A (projective, geometrically irreducible, non-singular) curve X defined over a finite field Fq2 is maximal if the number Nq2 of its Fq2-rational points attains the Hasse-Weil upper bound, that is Nq2=q2+2gq+1 where g is the genus of X. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order p2 where p is the characteristic of Fq2. Doing so we also determine the Fq2-isomorphism classes of such curves and describe their full Fq2-automorphism groups.

On the Dimensions of Hermitian Subfield Subcodes from Higher-Degree Places.

The focus of our research is the examination of Hermitian curves over finite fields, specifically concentrating on places of degree three and their role in constructing Hermitian codes. We begin by studying the structure of the Riemann-Roch space associated with these degree-three places, aiming to determine essential characteristics such as the basis. The investigation then turns to Hermitian codes, where we analyze both functional and differential codes of degree-three places, focusing on their parameters and automorphisms. In addition, we explore the study of subfield subcodes and trace codes, determining their structure by giving lower bounds for their dimensions. This presents a complex problem in coding theory. Based on numerical experiments, we formulate a conjecture for the dimension of some subfield subcodes of Hermitian codes. Our comprehensive exploration seeks to deepen the understanding of Hermitian codes and their associated subfield subcodes related to degree-three places, thus contributing to the advancement of algebraic coding theory and code-based cryptography.

Algebraic approach to the completeness problem for (k,n)-arcs in planes over finite fields

In a projective plane over a finite field, complete (k,n)-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the construction of small ones, with k close to the order of the plane, is considered a hard problem. A natural candidate to be a small (k,n)-arc with few characters is the set Ω(C) of the points of a plane curve C of degree n (containing no linear components) such that some line meets C transversally in the plane, i.e. in n pairwise distinct points. Let C be either the Hermitian curve of degree q+1 in PG(2,q2r) with r≥1, or the rational BKS curve of degree q+1 in PG(2,qr) with q odd and r≥1. Then Ω(C) has four and seven characters, respectively. Furthermore, Ω(C) is small as both curves are either maximal or minimal. The completeness problem is investigated by an algebraic approach based on Galois theory and on the Hasse-Weil lower bound. Our main result for the Hermitian case is that Ω(C) is complete for r≥4. For the rational BKS curve, Ω(C) is complete if and only if r is even. If r is odd then the uncovered points by the (q+1)-secants to Ω(C) are exactly the points in PG(2,q) not lying in Ω(C). Adding those points to Ω(C) produces a complete (k,q+1)-arc in PG(2,qr), with k=qr+q. The above results do not hold true for r=2 and there remain open the case r=3 for the Hermitian curve, and the cases r=3,4 for the rational BKS curve. As a by product we also obtain two results of interest in the study of the Galois inverse problem for PGL(2,q).

On maximal plane curves of degree $3$ over $\mathbb{F}_4$, and Sziklai's example of degree $q-1$ over $\mathbb{F}_q$

An elementary and self-contained argument for the complete determination of maximal plane curves of degree $3$ over $\mathbb{F}_4$ will be given, which complements Hirschfeld-Storme-Thas-Voloch's theorem on a characterization of Hermitian curves in $\mathbb{P}^2$. This complementary part should be understood as the classification of Sziklai's example of maximal plane curves of degree $q-1$ over $\mathbb{F}_q$. Although two maximal plane curves of degree $3$ over $\mathbb{F}_4$ up to projective equivalence over $\mathbb{F}_4$ appear, they are birationally equivalent over $\mathbb{F}_4$ each other.

Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus

In this article we explicitly determine the Weierstrass semigroup at any point and the full automorphism group of a known Fq2-maximal curve X3 having the third largest genus. This curve arises as a Galois subcover of the Hermitian curve, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough X3 has many different types of Weierstrass semigroups and the set of its Weierstrass points is much richer than its set of Fq2-rational points. This makes the curve X3 the first explicitly known maximal curve having non-rational Weierstrass points. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, Aut(X3) is exactly the automorphism group inherited from the Hermitian curve, apart from small values of q.

Fractional decoding of r-Hermitian codes

In this paper, we present a fractional decoding algorithm for a new family of codes which are constructed from the Hermitian curve, called r-Hermitian codes. These codes of length n are defined over an extension field Fq2l of Fq2 and the fractional decoding algorithms that we present are algorithms for error correction that use only αln symbols of a subfield of size q2 as input into the decoding algorithm, where α<1, meaning a fraction of the subsymbols that are typically utilized. We demonstrate that collaborative decoding of interleaved codes supports fractional decoding of the r-Hermitian codes, allowing for improved bounds on the fractional decoding radius.

Equations of motion governing the dynamics of the exceptional points of parameterically dependent nonhermitian Hamiltonians

We study exceptional points (EPs) of a nonhermitian Hamiltonian whose parameters and . As the real control parameter δ is varied, the kth EP (or kth cluster of simultaneously existing EPs) of moves in the complex plane of λ along a continuous trajectory, . Using an appropriate non-hermitian formalism (based upon the c-product and not upon the conventional Dirac product), we derive a self-contained set of equations of motion (EOM) for the trajectory , while interpreting δ as the propagation time. Such EOM become of interest whenever one wishes to study the response of EPs to external perturbations or continuous parametric changes of the pertinent Hamiltonian. This is e.g. the case of EPs emanating from hermitian curve crossings/degeneracies (which turn into avoided crossings/near-degeneracies when the Hamiltonian parameters are continuously varied). The presented EOM for EPs have not only their theoretical merits, they possess also a substantial practical relevance. Namely, the just presented approach can be regarded even as an efficient numerical method, useful for generating EPs for a broad class of complex quantum systems encountered in atomic, nuclear and condensed matter physics. Performance of such a method is tested here numerically on a simple yet nontrivial toy model.

Алгеброгеометрические коды и декодирование на основе пар, исправляющих ошибки

We consider the basic theory of algebraic curves and their function fields necessary for constructing algebraic geometry codes and a pair of codes forming an error-correction pair which is used in a precomputation step of the decoding algorithm for the algebraic geometry codes. Also, we consider the decoding algorithm and give the necessary theory to prove its correctness. As a result, we consider elliptic curves, Hermitian curves and Klein quartics and construct the algebraic geometry codes associated with these families of curves, and also explicitly define the error-correcting pairs for the resulting codes.

An algorithmic approach to entanglement-assisted quantum error-correcting codes from the Hermitian curve

&lt;p style='text-indent:20px;'&gt;We study entanglement-assisted quantum error-correcting codes (EAQECCs) arising from classical one-point algebraic geometry codes from the Hermitian curve with respect to the Hermitian inner product. Their only unknown parameter is &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ c $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, the number of required maximally entangled quantum states since the Hermitian dual of an AG code is unknown. In this article, we present an efficient algorithmic approach for computing &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ c $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for this family of EAQECCs. As a result, this algorithm allows us to provide EAQECCs with excellent parameters over any field size.&lt;/p&gt;

Generalized Weierstrass semigroups at several points on certain maximal curves which cannot be covered by the Hermitian curve

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 (J Pure Appl Algebra 220:1122–1132, 2016) as new examples of curves which cannot be covered by the Hermitian curve.

Classification of all Galois subcovers of the Skabelund maximal curves

In 2017 Skabelund constructed two new examples of maximal curves S˜q and R˜q as covers of the Suzuki and Ree curves, respectively. The resulting Skabelund curves are analogous to the Giulietti–Korchmáros cover of the Hermitian curve. In this paper a complete characterization of all Galois subcovers of the Skabelund curves S˜q and R˜q is given. Calculating the genera of the corresponding curves, we find new additions to the list of known genera of maximal curves over finite fields.

An elementary abelian p-cover of the Hermitian curve with many automorphisms

The full automorphism group of a certain elementary abelian p-cover of the Hermitian curve in characteristic $$p>0$$ is determined. It is remarkable that the order of Sylow p-groups of the automorphism group is close to Nakajima’s bound in terms of the p-rank. Weierstrass points, Galois points, Frobenius nonclassicality, and arc property are also investigated.

Hermitian-invariant codes from the Hermitian curve

Hermitian-invariant codes from the Hermitian curve

On algebraic curves with many automorphisms in characteristic p

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 \(g \ge 2\) then \(Aut({\mathcal {X}})\) is a finite group, and unless \({\mathcal {X}}\) is the so-called Hermitian curve, its order is upper bounded by a polynomial in g of degree four (Stichtenoth). In 1978 Henn proposed a refinement of Stichtenoth’s bound of degree 3 in g up to few exceptions, all having p-rank zero. In this paper a further refinement of Henn’s result is proposed. First, we prove that if an algebraic curve of genus \(g \ge 2\) has more than \(336g^2\) automorphisms then its automorphism group has exactly two short orbits, one tame and one non-tame, that is, the action of the group is completely known. Finally when \(|Aut({\mathcal {X}})| \ge 900g^2\) sufficient conditions for \({\mathcal {X}}\) to have p-rank zero are provided.

An 𝔽p2-maximal Wiman sextic and its automorphisms

AbstractIn 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equationX6+Y6+ℨ6+(X2+Y2+ℨ2)(X4+Y4+ℨ4)−12X2Y2ℨ2= 0, and showed that its full automorphism group is isomorphic to the symmetric groupS5. We show that this holds also over every algebraically closed field 𝕂 of characteristicp≥ 7. Forp= 2, 3 the above polynomial is reducible over 𝕂, and forp= 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman’s 𝔽192-maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19over the finite field 𝔽192.

𝔽p2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve

AbstractLet 𝔽 be the finite field of orderq2. It is sometimes attributed to Serre that any curve 𝔽-covered by the Hermitian curveHq+1:yq+1=xq+x${{\mathcal{H}}_{q+1}}:{{y}^{q+1}}={{x }^{q}}+x$is also 𝔽-maximal. For prime numbersqwe show that every 𝔽-maximal curvex$\mathcal{x}$of genusg≥ 2 with | Aut(𝒳) | &gt; 84(g− 1) is Galois-covered byHq+1.${{\mathcal{H}}_{q+1}}.$The hypothesis on | Aut(𝒳) | is sharp, since there exists an 𝔽-maximal curvex$\mathcal{x}$forq= 71 of genusg= 7 with | Aut(𝒳) | = 84(7 − 1) which is not Galois-covered by the Hermitian curveH72.${{\mathcal{H}}_{72}}.$

Hermitian-lifted codes

In this paper, we construct codes for local recovery of erasures with high availability and constant-bounded rate from the Hermitian curve. These new codes, called Hermitian-lifted codes, are evaluation codes with evaluation set being the set of $${{\mathbb {F}}}_{q^2}$$ -rational points on the affine curve. The novelty is in terms of the functions to be evaluated; they are a special set of monomials which restrict to low degree polynomials on lines intersected with the Hermitian curve. As a result, the positions corresponding to points on any line through a given point act as a recovery set for the position corresponding to that point.

Triples of rational points on the Hermitian curve and their Weierstrass semigroups

In this paper, we study configurations of three rational points on the Hermitian curve over F q 2 and classify them according to their Weierstrass semigroups . For q > 3 , we show that the number of distinct semigroups of this form is equal to the number of positive divisors of q + 1 and give an explicit description of the Weierstrass semigroup for each triple of points studied. To do so, we make use of two-point discrepancies and derive a criterion which applies to arbitrary curves over a finite field.