arXiv subjects

Alonso S. Castellanos

Publications and source records attributed to Alonso S. Castellanos.

13 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↗

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↗

arXiv2024

Linear Complementary dual codes and Linear Complementary pairs of AG codes in function fields

In recent years, linear complementary pairs (LCP) of codes and linear complementary dual (LCD) codes have gained significant attention due to their applications in coding theory and cryptography. In this work, we construct explicit LCPs of codes and LCD codes from function fields of genus $g \geq 1$. To accomplish this, we present pairs of suitable divisors giving rise to non-special divisors of degree $g-1$ in the function field. The results are applied in constructing LCPs of algebraic geometry codes and LCD algebraic geometry (AG) codes in Kummer extensions, hyperelliptic function fields, and elliptic curves.

math.AG↗

arXiv2024

Self-orthogonal flags of codes and translation of flags of algebraic geometry codes

A flag $C_0 \subsetneq C_1 \cdots \subsetneq C_s \subsetneq {\mathbb F}_q^n $ of linear codes is said to be self-orthogonal if the duals of the codes in the flag satisfy $C_{i}^\perp=C_{s-i}$, and it is said to satisfy the isometry-dual property with respect to an isometry vector ${\bf x}$ if $C_i^\perp={\bf x} C_{s-i}$ for $i=1, \dots, s$. We characterize complete (i.e. $s=n$) flags with the isometry-dual property by means of the existence of a word with non-zero coordinates in a certain linear subspace of ${\mathbb F}_q^n$. For flags of algebraic geometry (AG) codes we prove a so-called translation property of isometry-dual flags and give a construction of complete self-orthogonal flags, providing examples of self-orthogonal flags over some maximal function fields. At the end we characterize the divisors giving the isometry-dual property and the related isometry vectors showing that for each function field there is only a finite number of isometry vectors and that they are related by cyclic repetitions.

cs.IT↗

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

Isometry-Dual Flags of Many-Point AG Codes

Let $F_q$ be a finite field. A flag of $F_q$-linear codes $C_0\subsetneq C_1\subsetneq\dots\subsetneq C_s$ is said to satisfy the isometry-dual property if there exists a vector $x\in(F_q^*)^n$ such that $C_i=x\cdot C_{s-i}^\perp$, where $C_i^\perp$ denotes the dual code of $C_i$. Consider $F/F_q$ a function field and let $P$ and $Q_1,\ldots,Q_t$ be rational places of $F$. Let the divisor $D$ be the sum of pairwise different places of $F$ such that $P, Q_1,\dots,Q_t$ are not in $supp(D)$. In a previous work we investigated the existence of flags of two-point codes $C(D,a_0P+bQ_1)\subsetneq C(D,a_1P+bQ_1))\subsetneq\dots\subsetneq C(D,a_sP+bQ_1)$ satisfying the isometry-dual property for a non-negative integer $b$ and an increasing sequence of positive integers $a_0,\dots,a_s$. While for one-point codes (i.e. for $b=0$) there is only need to analyze positive integers $a$, for the case of $(t+1)$-point codes, the integers $a$ may be negative. We extend our previous results in different directions. On one hand to the case of negative integers $a$ and $b$, and on the other hand we extend our results to flags of $(t+1)$-point codes $C(D,a_0P+\sum_{i=1}^tβ_iQ_i)\subsetneq C(D, a_1P+\sum_{i=1}^tβ_iQ_i))\subsetneq\dots\subsetneq C(D, a_sP+\sum_{i=1}^tβ_iQ_i)$ for any tuple of (either positive or negative) integers $β_1,\dots,β_t$ and for an increasing sequence of (either positive or negative) integers $a_0,\dots,a_s$. We apply the obtained results to the broad class of Kummer extensions defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $gcd(r, m)=1$. In particular, depending on the place $P$ and for $D$ an $Aut(F_q(x, y)/F_q(x))$-invariant sum of rational places of $F$ such that $P,Q_i\notin supp(D)$, we obtain necessary and sufficient conditions on $m$ and $β_i$'s such that the flag has the isometry-dual property.

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

Weierstrass Semigroup, Pure Gaps and Codes on Function Fields

We determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation $y^{m}=\prod_{i=1}^{r} (x-α_i)^{λ_i}$ over $K$, the algebraic closure of $\mathbb{F}_q$, where $α_1, \dots, α_r\in K$ are pairwise distinct elements, and $\gcd(m, \sum_{i=1}^{r}λ_i)=1$. For an arbitrary function field, from the knowledge of the minimal generating set of the Weierstrass semigroup at two rational places, the set of pure gaps is characterized. We apply these results to construct algebraic geometry codes over certain function fields with many rational places.

math.AG↗

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↗

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↗

arXiv2021

The Isometry-Dual Property in Flags of Two-Point Algebraic Geometry Codes

A flag of codes $C_0 \subsetneq C_1 \subsetneq \cdots \subsetneq C_s \subseteq {\mathbb F}_q^n$ is said to satisfy the {\it isometry-dual property} if there exists ${\bf x}\in (\mathbb{F}_q^*)^n$ such that the code $C_i$ is {\bf x}-isometric to the dual code $C_{s-i}^\perp$ for all $i=0,\ldots, s$. For $P$ and $Q$ rational places in a function field ${\mathcal F}$, we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes $$C_\mathcal L(D, a_0P+bQ)\subsetneq C_\mathcal L(D, a_1P+bQ)\subsetneq \dots \subsetneq C_\mathcal L(D, a_sP+bQ),$$ where the divisor $D$ is the sum of pairwise different rational places of ${\mathcal F}$ and $P, Q$ are not in $\mbox{supp}(D)$. We characterize those sequences in terms of $b$ for general function fields. We then apply the result to the broad class of Kummer extensions ${\mathcal F}$ defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $\mbox{gcd}(r, m)=1$. For $P$ the rational place at infinity and $Q$ the rational place associated to one of the roots of $f(x)$, it is shown that the flag of two-point algebraic geometry codes has the isometry-dual property if and only if $m$ divides $2b+1$. At the end we illustrate our results by applying them to two-point codes over several well know function fields.

cs.IT↗

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

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↗