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Helena Martín-Cruz

Publications and source records attributed to Helena Martín-Cruz.

8 recordsLinked to original sources
arXiv2026

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

Quantum $(r,δ)$-locally recoverable codes are quantum error-correcting codes capable of correcting $δ-1$ qudit erasures within one subset of qudits of cardinality at most $r+δ-1$. In this paper, we introduce the more general framework of entanglement-assisted quantum $(r,δ)$-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 $(r,δ)$-local recoverability, and derive a Singleton-like bound. Furthermore, we construct optimal pure entan\-gle\-ment-assisted quantum $(r,δ)$-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↗
arXiv2025

Quantum $(r,δ)$-locally recoverable codes

Classical $(r,δ)$-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 $(r,δ)$-locally recoverable codes which are quantum error-correcting codes capable of correcting $δ-1$ qudit erasures from sets of at most $r+ δ-1$ qudits. We give a necessary and sufficient condition for a quantum stabilizer code $Q(C)$ to be $(r,δ)$-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 $(r,δ)$-local recoverability. A Singleton-like bound is stated in this case and examples attaining the bound are given.

cs.IT↗
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 $λn_1$, where $λ$ and $n_1$ are suitable positive integers such that the punctured code $\mathcal{B}$ of $\mathcal{C}$ in the last $λn_1 - n_1$ 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↗
arXiv2023

MDS, Hermitian Almost MDS, and Gilbert-Varshamov Quantum Codes from Generalized Monomial-Cartesian Codes

We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials in $m$ variables. When $m=1$ our codes are MDS, and when $m=2$ and our lower bound for the minimum distance is $3$ the codes are at least Hermitian Almost MDS. For an infinite family of parameters when $m=2$ we prove that our codes beat the Gilbert-Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.

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↗
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↗