arXiv subjects

Carlos Galindo

Publications and source records attributed to Carlos Galindo.

At least 19 recordsLinked to original sources

arXiv2026

Entanglement assisted quantum $(r,\delta)$-locally recoverable codes

Quantum $(r,\delta)$-locally recoverable codes are quantum error-correcting codes capable of correcting $\delta-1$ qudit erasures within one subset of qudits of cardinality at most $r+\delta-1$. In this paper, we introduce the more general framework of entanglement-assisted quantum $(r,\delta)$-locally recoverable codes, assuming that the local recovery operation is assisted by receiver-held qudits that remain unaffected by erasures. We establish necessary and sufficient conditions for these codes to satisfy this property. For codes derived from Hermitian or Euclidean constructions, we establish connections between entanglement-assisted quantum and classical notions of $(r,\delta)$-local recoverability, and derive a Singleton-like bound. Furthermore, we construct optimal pure entan\-gle\-ment-assisted quantum $(r,\delta)$-locally recoverable codes from several families of classical codes, including bivariate $J$-affine variety codes, BCH codes, and homothetic-BCH codes.

quant-ph↗

arXiv2026

Impure codes exceeding the pure bounds for quantum local recovery

Existing literature provides several bounds for quantum local recovery, which essentially consider the number of message qudits, the distance, the length, and the locality of the involved codes. We give a family of $J$-affine variety codes that result in impure CSS codes. These quantum codes exceed several of the above mentioned bounds that apply to pure quantum locally recoverable codes. We also discuss a connection between bounds on quantum local recovery and on weight-constrained stabilizer codes.

cs.IT↗

arXiv2026

Nef divisors of surfaces given by pencils at infinity

We give generators for the nef cone and the cone of curves of rational surfaces obtained by blowing-up the complex projective plane at a set of points $\mathcal{B} \cup \mathcal{D}$, where $\mathcal{B}$ is the set of (proper and infinitely near) base points of a pencil associated with a curve having one place at infinity, and $\mathcal{D}$ is a set of finitely many infinitely near free points on the strict transforms of curves of the pencil. We also prove that, when the pencil is given by an AMS-type curve and $\mathcal{D}$ contains at most two free points on any curve considered, the Cox ring of the obtained surface is finitely generated.

math.AG↗

arXiv2026

Quantum $(r,\delta)$-Locally Recoverable BCH and Homothetic-BCH Codes

Quantum $(r,\delta)$-locally recoverable codes ($(r,\delta)$-LRCs) are the quantum version of classical $(r,\delta)$-LRCs designed to recover multiple failures in large-scale distributed and cloud storage systems. A quantum $(r,\delta)$-LRC, $Q(C)$, can be constructed from an $(r,\delta)$-LRC, $C$, which is Euclidean or Hermitian dual-containing. This article is devoted to studying how to get quantum $(r,\delta)$-LRCs from BCH and homothetic-BCH codes. As a consequence, we give pure quantum $(r,\delta)$-LRCs which are optimal for the Singleton-like bound.

cs.IT↗

arXiv2025

The cone of curves and the Cox ring of rational surfaces over Hirzebruch surfaces

Let $X$ be a rational surface obtained by blowing up at a configuration $\mathcal{C}$ of infinitely near points over a Hirzebruch surface $\mathbb{F}_δ$. We prove that there exist two positive integers $a \leq b$ such that the cone of curves of $X$ is finite polyhedral and minimally generated when $δ\geq a$, and the Cox ring of $X$ is finitely generated whenever $δ\geq b$. The integers $a$ and $b$ depend only on a combinatorial object (a graph decorated with arrows) representing the strict transforms of the exceptional divisors, their intersections and those with the fibers and special section of $\mathbb{F}_δ$.

math.AG↗

arXiv2025

New quantum codes from homothetic-BCH codes

We introduce homothetic-BCH codes. These are a family of $q^2$-ary classical codes $\mathcal{C}$ of length $\lambda n_1$, where $\lambda$ and $n_1$ are suitable positive integers such that the punctured code $\mathcal{B}$ of $\mathcal{C}$ in the last $\lambda n_1 - n_1$ coordinates is a narrow-sense BCH code of length $n_1$. We prove that whenever $\mathcal{B}$ is Hermitian self-orthogonal, so is $\mathcal{C}$. As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths than cannot be reached by BCH codes. With this procedure we get new quantum codes according to Grassl's table. To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths.

cs.IT↗

arXiv2025

Linear weighted bounded negativity

We propose a linear version of the weighted bounded negativity conjecture. It considers a smooth projective surface $X$ over an algebraically closed field of characteristic zero and predicts the existence of a common lower bound on $C^2/(D\cdot C)$ for all reduced and irreducible curves $C$ and all big and nef divisors such that $D\cdot C>0$, both on $X$. We prove that, in the complex case, there exists such a bound for all nef divisors spanning a ray out an open covering of the limit rays of negative curves. In the same vein, we provide explicit bounds when $X$ is a rational surface. Our proofs involve the existence of a foliation $\mathcal{F}$ on $X$ but most of our results are independent of $\mathcal{F}$.

math.AG↗

arXiv2024

Quantum $(r,\delta)$-locally recoverable codes

Classical $(r,\delta)$-locally recoverable codes are designed for avoiding loss of information in large scale distributed and cloud storage systems. We introduce the quantum counterpart of those codes by defining quantum $(r,\delta)$-locally recoverable codes which are quantum error-correcting codes capable of correcting $\delta -1$ qudit erasures from sets of at most $r+ \delta -1$ qudits. We give a necessary and sufficient condition for a quantum stabilizer code $Q(C)$ to be $(r,\delta)$-locally recoverable. Our condition depends only on the puncturing and shortening at suitable sets of both the symplectic self-orthogonal code $C$ used for constructing $Q(C)$ and its symplectic dual $C^{\perp_s}$. When $Q(C)$ comes from a Hermitian or Euclidean dual-containing code, and under an extra condition, we show that there is an equivalence between the classical and quantum concepts of $(r,\delta)$-local recoverability. A Singleton-like bound is stated in this case and examples attaining the bound are given.

cs.IT↗

arXiv2024

On weighted bounded negativity for rational surfaces

The weighted bounded negativity conjecture considers a smooth projective surface $X$ and looks for a common lower bound on the quotients $C^2/(D\cdot C)^2$, where $C$ runs over the integral curves on $X$ and $D$ over the big and nef divisors on $X$ such that $D \cdot C >0$. We focus our study on rational surfaces $Z$. Setting $π: Z \rightarrow Z_0$ a composition of blowups giving rise to $Z$, where $Z_0$ is the projective plane or a Hirzebruch surface, we give a common lower bound on $C^2/(H^* \cdot C)^2$ whenever $H^*$ is the pull-back of a nef divisor $H$ on $Z_0$. In addition, we prove that, only in the case when a nef divisor $D$ on $Z$ approaches the boundary of the nef cone, the quotients $C^2/(D\cdot C)^2$ could tend to minus infinity.

math.AG↗

arXiv2024

Algebraic integrability with bounded genus

We provide an algorithm which decides whether a polynomial foliation $\mathcal{F}^{\mathbb{C}^2}$ on the complex plane has a polynomial first integral of genus $g\neq 1$. Except in a specific case, an extension of the algorithm also decides if $\mathcal{F}^{\mathbb{C}^2}$ has a rational first integral of that genus.

math.AG↗

arXiv2023

Optimal pure quantum $(r,\delta)$-locally recoverable codes from matrix-product construction

Locally recoverable codes (LRCs) are classical error-correcting codes widely used in large-scale distributed and cloud storage systems. Quantum locally recoverable codes of locality $(r,\delta)$ (quantum $(r,\delta)$-LRCs) are the quantum counterpart of classical $(r,\delta)$-LRCs. They allow us to correct erasures at several positions using a trace-preserving quantum operation acting on qudits of a larger set of positions. Quantum $(r,\delta)$-LRCs, $\mathcal{Q}(\mathcal{C})$, can be constructed from classical Euclidean (or Hermitian) dual-containing codes $\mathcal{C}$, and their recovery abilities are upper bounded by the minimum distance of the Euclidean (or Hermitian) dual of those codes. Parameters and localities of pure quantum $(r,\delta)$-LRCs satisfy a Singleton-like bound; codes attaining equality are referred to as optimal. We consider matrix-product codes (MPCs) $\mathcal{C}$ and give constituent (or defining) matrices and conditions on the constituent codes such that the codes $\mathcal{C}$ satisfy the conditions to provide quantum $(r,\delta)$-LRCs. As a consequence, we are able to determine their locality and parameters. Furthermore, we determine families of optimal pure quantum $(r,\delta)$-LRCs derived from them.

cs.IT↗

arXiv2023

Optimal $(r,δ)$-LRCs from monomial-Cartesian codes and their subfield-subcodes

We study monomial-Cartesian codes (MCCs) which can be regarded as $(r,δ)$-locally recoverable codes (LRCs). These codes come with a natural bound for their minimum distance and we determine those giving rise to $(r,δ)$-optimal LRCs for that distance, which are in fact $(r,δ)$-optimal. A large subfamily of MCCs admits subfield-subcodes with the same parameters of certain optimal MCCs but over smaller supporting fields. This fact allows us to determine infinitely many sets of new $(r,δ)$-optimal LRCs and their parameters.

cs.IT↗

arXiv2023

Steane enlargement of Entanglement-Assisted Quantum Error-Correcting Codes

We introduce a Steane-like enlargement procedure for entanglement-assisted quantum error-correcting codes (EAQECCs) obtained by considering Euclidean inner product. We give formulae for the parameters of these enlarged codes and apply our results to explicitly compute the parameters of enlarged EAQECCs coming from some BCH codes.

cs.IT↗

arXiv2022

On the valuative Nagata conjecture

We provide several equivalent conditions for a plane divisorial valuation of a smooth projective surface to be minimal with respect to an ample divisor. These conditions involve a valuative Seshadri constant and other global tools of the surface defined by the divisorial valuation. As a consequence, we derive several equivalent statements for the valuative Nagata conjecture and some related results.

math.AG↗

arXiv2022

Stabilizer quantum codes defined by trace-depending polynomials

Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a constant plus the trace of a polynomial) and show that this procedure gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in other papers involving roots of the trace and with excellent parameters. Namely, we are able to provide new binary records and non-binary codes improving the ones available in the literature.

cs.IT↗

arXiv2022

On the degree of curves with prescribed multiplicities and bounded negativity

We provide a lower bound on the degree of curves of the projective plane $\mathbb{P}^2$ passing through the centers of a divisorial valuation $ν$ of $\mathbb{P}^2$ with prescribed multiplicities, and an upper bound for the Seshadri-type constant of $ν$, $\hatμ(ν)$, constant that is crucial in the Nagata-type valuative conjecture. We also give some results related to the bounded negativity conjecture concerning those rational surfaces having the projective plane as a relatively minimal model.

math.AG↗

arXiv2022

Algebraic integrability of planar polynomial vector fields by extension to Hirzebruch surfaces

We study algebraic integrability of complex planar polynomial vector fields $X=A (x,y)(\partial/\partial x) + B(x,y) (\partial/\partial y) $ through extensions to Hirzebruch surfaces. Using these extensions, each vector field $X$ determines two infinite families of planar vector fields that depend on a natural parameter which, when $X$ has a rational first integral, satisfy strong properties about the dicriticity of the points at the line $x=0$ and of the origin. As a consequence, we obtain new necessary conditions for algebraic integrability of planar vector fields and, if $X$ has a rational first integral, we provide a region in $\mathbb{R}_{\geq 0}^2$ that contains all the pairs $(i,j)$ corresponding to monomials $x^i y^j$ involved in the generic invariant curve of $X$.

math.AG↗

arXiv2021

On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes

Many $q$-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal $q^2$-ary linear codes. This result can be generalized to $q^{2 m}$-ary linear codes, $m > 1$. We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to \cite{codet} and new $q$-ary ones, with $q \neq 2$, improving others in the literature.

cs.IT↗