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Guilherme Tizziotti

Publications and source records attributed to Guilherme Tizziotti.

18 recordsLinked to original sources
arXiv2026

Permutation Decoding of AG Codes from Curves Defined by Separated Polynomials

In this work, we investigate permutation decoding for algebraic geometry (AG) codes arising from algebraic curves defined by separated polynomials. Using automorphisms of the underlying curves, we construct permutation automorphisms of the associated algebraic geometry codes and exploit the resulting orbit structure to determine information and check positions. We introduce a class of curves, called SAP curves (Separated Additive Polynomial curves), and investigate one-point AG codes defined on them. For these codes, we obtain permutation decoding sets that correct burst errors supported on coordinates associated with rational points sharing a common coordinate. We further identify a subclass of special SAP curves, including Hermitian curves, generalized Hermitian curves, and certain maximal curves, for which additional automorphisms yield more powerful decoding sets.

cs.IT↗
arXiv2026

A closed formula for the Geil-Matsumoto bound on numerical semigroups via Apéry sets

The Geil-Matsumoto bound (GM bound) constrains the number of rational points on a curve over a finite field in terms of the Weierstrass semigroup of any of the points on the curve. For general numerical semigroups, the GM bound lacks a simple closed-form expression, making its computation a challenging problem. A closed formula has been obtained for the case when the semigroup is generated by two co-prime integers. In this work, for any numerical semigroup, we provide a closed formula for the GM bound in terms of the Apéry set of a nonzero element of the semigroup. In the case where the numerical semigroup is generated by consecutive integers $n, n+1, \dots, n+t$ with $\lceil\textstyle\frac{n-1}{2}\rceil\leq t \leq n-1$, we obtain a simple closed formula for the bound. We apply these results to obtain upper bounds on the number of rational points for algebraic curves over finite fields. In some cases, our bounds improve some well-known upper bounds on the number of rational points.

math.NT↗
arXiv2026

On planar functions over $\mathbb{F}_{q^3}$

Let $\mathbb{F}_q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over $\mathbb{F}_{q^3}$ of the family $$ f_{E,A,B,C,D}(X) := EX^2+ AX^{q+1}+ BX^{q^2+1}+CX^{2q} +DX^{2q^2}\in \mathbb{F}_{q}[X]. $$ Using results from the theory of q-polynomials, we establish conditions under which these polynomials are planar functions. In particular, we provide characterizations for the planarity property and present new families of planar trinomials, quadrinomials, and pentanomials.

math.NT↗
arXiv2025

On generalized Weierstrass Semigroups in arbitrary Kummer extensions of $\mathbb{F}_q(x)$

In this work, we investigate generalized Weierstrass semigroups in arbitrary Kummer extensions of function field $\mathbb{F}_q(x)$. We analyze their structure and properties, with a particular emphasis on their maximal elements. Explicit descriptions of the sets of absolute and relative maximal elements within these semigroups are provided. Additionally, we apply our results to function fields of the maximal curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$, which cannot be covered by the Hermitian curve, and the Beelen-Montanucci curve. Our results generalize and unify several earlier contributions in the theory of Weierstrass semigroups, providing new perspectives on the relationship between these semigroups and function fields.

math.AG↗
arXiv2023

The Set of Pure Gaps at Several Rational Places in Function Fields

In this work, using maximal elements in generalized Weierstrass semigroups and its relationship with pure gaps, we extend the results in \cite{CMT2024} and provide a way to completely determine the set of pure gaps at several rational places in an arbitrary function field $F$ over a finite field and its cardinality. As an example, we determine the cardinality and a simple explicit description of the set of pure gaps at several rational places distinct to the infinity place on Kummer extensions, which is a different characterization from that presented by Hu and Yang in \cite{HY2018}. Furthermore, we present some applications in coding theory and AG codes with good parameters.

cs.IT↗
arXiv2023

On atoms of the set of generalized numerical semigroups with fixed corner element

We study the so-called atomic GNS, which naturally extends the concept of atomic numerical semigroup. We introduce the notion of corner special gap and we characterize the class of atomic GNS in terms of the cardinality of the set of corner special gaps and also in terms of a maximal property. Using this maximal property we present some properties concerning irreducibility of Frobenius GNSs. In particular, we provide sufficient conditions for certain Frobenius GNSs to be an atom non-irreducible (ANI). Furthermore, we given necessary and sufficient conditions so that the maximal elements of a set of Frobenius GNSs with two fixes gaps to be all irreducible or not.

math.GR↗
arXiv2023

Complete set of Pure Gaps in Function Fields

In this work, we provide a way to completely determine the set of pure gaps $G_0(P_1, P_2)$ at two rational places $P_1, P_2$ in a function field $F$ over a finite field $\mathbb{F}_q$, and its cardinality. Furthermore, we given a bound for the cardinality of the set $G_0(P_1, P_2)$ which is better, in some cases, than the generic bound given by Homma and Kim. As a consequence, we completely determine the set of pure gaps and its cardinality for two families of function fields: the $GK$ function field and Kummer extensions.

math.NT↗
arXiv2023

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 $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$ introduced in Tafazolian et al. in 2016 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.

math.AG↗
arXiv2022

Construction of sequences with high nonlinear complexity from a generalization of the Hermitian function field

For $r \geq 1$ an odd integer, we provide a sequence from the function field $\mathcal{F}_{q, r}$ of the maximal curve over $\mathbb{F}_{q^{2r}}$ defined by the affine equation $y^q+y=x^{q^r + 1}$. This sequence has high nonlinear complexity, and this fact comes from the existence of a rational function on $\mathcal{F}_{q, r}$ with pole divisor of small degree, and support in certain $q$ rational places.

cs.IT↗
arXiv2022

The corner element of generalized numerical semigroups

In this paper we introduce the concept of corner element of a generalized numerical semigroup, which extends in a sense the idea of conductor of a numerical semigroup to generalized numerical semigroups in higher dimensions. We present properties of this new notion and its relations with existing invariants in the literature, and provide an algorithm to compute all the generalized numerical semigroups with fixed corner. Besides that, we provide lower and upper bounds on the number of generalized numerical semigroups having a fixed corner element.

math.GR↗
arXiv2021

On the existence of pairs of primitive and normal elements over finite fields

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) \in \mathbb{F}_q[x]$ with $\textrm{deg}(f_i(x)) \leq m_i$, for $i = 1, 2$, and such that the rational function $f_1(x)/f_2(x)$ belongs to a certain set which we define, we present a sufficient condition for the existence of a primitive element $α\in \mathbb{F}_{q^n}$, normal over $\mathbb{F}_q$, such that $f_1(α)/f_2(α)$ is also primitive.

math.NT↗
arXiv2018

On Weierstrass gaps at several points

We consider the problem of determining Weierstrass gaps and pure Weierstrass gaps at several points. Using the notion of relative maximality in generalized Weierstrass semigroups due to Delgado \cite{D}, we present a description of these elements which generalizes the approach of Homma and Kim \cite{HK} given for pairs. Through this description, we study the gaps and pure gaps at several points on a certain family of curves with separated variables.

math.AG↗
arXiv2018

Subcovers and codes on the $X_{n,r}$ curves

In this work, subcovers $\mathcal{X}_{n,r}^s$ of the curve $\mathcal{X}_{n,r}$ are constructed, the Weierstrass semigroup $H(P_\infty)$ at the point $P_\infty \in \mathcal{X}_{n,r}^s$ is determined and the corresponding one-point AG codes are investigated. Codes establishing new records on the parameters with respect to the previously known ones are discovered, and $108$ improvements on MinT tables are obtained.

math.AG↗
arXiv2017

Generalized Weierstrass semigroups and Riemann-Roch spaces for certain curves with separated variables

In this work we study the generalized Weierstrass semigroup $\widehat{H} (\mathbf{P}_m)$ at an $m$-tuple $\mathbf{P}_m = (P_{1}, \ldots , P_{m})$ of rational points on certain curves admitting a plane model of the form $f(y) = g(x)$ over $\mathbb{F}_{q}$, where ${f(T),g(T)\in \mathbb{F}_q[T]}$. In particular, we compute the generating set $\widehatΓ(\mathbf{P}_m)$ of $\widehat{H} (\mathbf{P}_m)$ and, as a consequence, we explicit a basis for Riemann-Roch spaces of divisors with support in $\{P_{1}, \ldots , P_{m}\}$ on these curves, generalizing results of Maharaj, Matthews, and Pirsic.

math.AG↗
arXiv2017

Weierstrass Semigroup and Pure Gaps at several points on the $GK$ curve

We determine the Weierstrass semigroup $H(P_{\infty}, P_{1}, \ldots , P_{m})$ at several points on the $GK$ curve. In addition, we present conditions to find pure gaps on the set of gaps $G(P_{\infty}, P_{1}, \ldots , P_{m})$. Finally, we apply the results to obtain AG codes with good relative parameters.

math.AG↗
arXiv2017

On Gröbner Basis for certain one-point AG codes

In this work we present a way to construct the so-called root diagram for one-point AG codes $C$ arising from certain types of curves $\mathcal{X}$ over $\mathbb{F}_q$ with plane model $f(y)=g(x)$. Using this root diagram we can get an algorithm to obtain a Gröbner basis for the submodule $\overline{C}$ associated to $C$

math.AG↗
arXiv2015

Two-point AG codes on the GK maximal curves

We determine de Weierstrass semigroup of a pair of certain rational points on the GK-curves. We use this semigroup to obtain two-point AG codes with better parameters than comparable one-point AG codes arising from these curves. These parameters are new records in the MinT's tables.

math.AG↗