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Diego Ruano

Publications and source records attributed to Diego Ruano.

At least 19 recordsLinked to original sources
arXiv2026

Quantum Matrix-Product Codes: CSS-T Characterization and Maximality

CSS-T codes are quantum error-correcting codes that play an important role in fault-tolerant quantum computation, as they help mitigate the proliferation of errors. They admit a transversal T-gate and are defined from a pair of nested classical binary linear codes satisfying a specific algebraic condition expressed in terms of their Schur square. We extend this algebraic characterization to the propagation rule known as the $(u \mid u+v)$-construction, that is, the matrix-product code constructed from two constituent codes. Moreover, for cyclic constituent codes, we provide an explicit characterization in terms of the defining cyclotomic sets, which extends existing results for cyclic codes. This framework allows the construction of new and longer CSS-T codes.

cs.IT↗
arXiv2025

A combinatorial description of when a self-associated set of points fails to be arithmetically Gorenstein

We prove that the set of points associated to a self-dual code with no proportional columns is arithmetically Gorenstein if and only if the code is indecomposable. This answers a question asked by Toh{\u{a}}neanu. We do so by providing a combinatorial way to compute the dimension of the Schur square of a self-dual code through a zero-one symmetrization of its generator matrix. Our approach also allows us to compute the Gorenstein defect. As a consequence, we obtain a combinatorial characterization of arithmetically Gorenstein self-associated sets of points over an algebraically closed field.

math.CO↗
arXiv2025

Duals of multiplicity codes

Multivariate multiplicity codes have been recently explored because of their importance for list decoding and local decoding. Given a multivariate multiplicity code, in this paper, we compute its dimension using Gr\"obner basis tools, its dual in terms of indicator functions, and explicitly describe a parity-check matrix. In contrast with Reed--Muller, Reed--Solomon, univariate multiplicity, and other evaluation codes, the dual of a multivariate multiplicity code is not equivalent or isometric to a multiplicity code (i.e., this code family is not closed under duality). We use our explicit description to provide a lower bound on the minimum distance for the dual of a multiplicity code.

cs.IT↗
arXiv2025

The Schur product of evaluation codes and its application to CSS-T quantum codes and private information retrieval

In this work, we study the componentwise (Schur) product of monomial-Cartesian codes by exploiting its correspondence with the Minkowski sum of their defining exponent sets. We show that $ J$-affine variety codes are well suited for such products, generalizing earlier results for cyclic, Reed-Muller, hyperbolic, and toric codes. Using this correspondence, we construct CSS-T quantum codes from weighted Reed-Muller codes and from binary subfield-subcodes of $ J$-affine variety codes, leading to codes with better parameters than previously known. Finally, we present Private Information Retrieval (PIR) constructions for multiple colluding servers based on hyperbolic codes and subfield-subcodes of $ J$-affine variety codes, and show that they outperform existing PIR schemes.

cs.IT↗
arXiv2024

Binary Triorthogonal and CSS-T Codes for Quantum Error Correction

In this paper, we study binary triorthogonal codes and their relation to CSS-T quantum codes. We characterize the binary triorthogonal codes that are minimal or maximal with respect to the CSS-T poset, and we also study how to derive new triorthogonal matrices from existing ones. Given a binary triorthogonal matrix, we characterize which of its equivalent matrices are also triorthogonal. As a consequence, we show that a binary triorthogonal matrix uniquely determines the parameters of the corresponding triorthogonal quantum code, meaning that any other equivalent matrix that is also triorthogonal gives rise to a triorthogonal quantum code with the same parameters.

cs.IT↗
arXiv2023

An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance

CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes $(C_1, C_2)$ such that $C_1$ contains $C_2$, $C_2$ is even, and the shortening of the dual of $C_1$ with respect to the support of each codeword of $C_2$ is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes $(C_1, C_2)$ is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed-Muller, cyclic, and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.

cs.IT↗
arXiv2023

Quantum error-correcting codes from projective Reed-Muller codes and their hull variation problem

Long quantum codes using projective Reed-Muller codes are constructed. Projective Reed-Muller codes are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. We provide entanglement-assisted quantum error-correcting codes from projective Reed-Muller codes with flexible amounts of entanglement by considering equivalent codes. Moreover, we also construct quantum codes from subfield subcodes of projective Reed-Muller codes.

cs.IT↗
arXiv2023

Hulls of projective Reed-Muller codes over the projective plane

By solving a problem regarding polynomials in a quotient ring, we obtain the relative hull and the Hermitian hull of projective Reed-Muller codes over the projective plane. The dimension of the hull determines the minimum number of maximally entangled pairs required for the corresponding entanglement-assisted quantum error-correcting code. Hence, by computing the dimension of the hull we now have all the parameters of the symmetric and asymmetric entanglement-assisted quantum error-correcting codes constructed with projective Reed-Muller codes over the projective plane. As a byproduct, we also compute the dimension of the Hermitian hull for affine Reed-Muller codes in 2 variables.

cs.IT↗
arXiv2023

Single Server Private Information Retrieval Protocols With Codes Over Rings

A Private Information Retrieval (PIR) protocol based on coding theory for a single server is proposed. It provides computational security against linear algebra attacks, addressing the main drawback of previous PIR proposals based on coding theory. The approach involves two types of codes each one over a different ring, an inner non-free linear code that will be used as a distinguisher of some elements added to the query matrix, and an outer code that will be used for generating the query matrix. Moreover, it only uses modular arithmetic at the server level and the recovering stage if the base ring chosen for the inner code is $\mathbb Z_m$.

cs.IT↗
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

Subfield subcodes of projective Reed-Muller codes

Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective space, we generalize the necessary tools to deal with this case as well: we obtain a universal Gr\"obner basis for the vanishing ideal of the set of standard representatives of the projective space and we show how to reduce any monomial with respect to this Gr\"obner basis. With respect to the parameters of these codes, by considering subfield subcodes of projective Reed-Muller codes we obtain long linear codes with good parameters over a small finite field.

cs.IT↗
arXiv2022

Relative hulls and quantum codes

Given two $q$-ary codes $C_1$ and $C_2$, the relative hull of $C_1$ with respect to $C_2$ is the intersection $C_1\cap C_2^\perp$. We prove that when $q>2$, the relative hull dimension can be repeatedly reduced by one, down to a certain bound, by replacing either of the two codes with an equivalent one. The reduction of the relative hull dimension applies to hulls taken with respect to the $e$-Galois inner product, which has as special cases both the Euclidean and Hermitian inner products. We give conditions under which the relative hull dimension can be increased by one via equivalent codes when $q>2$. We study some consequences of the relative hull properties on entanglement-assisted quantum error-correcting codes and prove the existence of new entanglement-assisted quantum error-correcting maximum distance separable codes, meaning those whose parameters satisfy the quantum Singleton bound.

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↗
arXiv2022

Saturation and vanishing ideals

We consider an homogeneous ideal $I$ in the polynomial ring $S=K[x_1,\dots,$ $x_m]$ over a finite field $K=\mathbb{F}_q$ and the finite set of projective rational points $\mathbb{X}$ that it defines in the projective space $\mathbb{P}^{m-1}$. We concern ourselves with the problem of computing the vanishing ideal $I(\mathbb{X})$. This is usually done by adding the equations of the projective space $I(\mathbb{P}^{m-1})$ to $I$ and computing the radical. We give an alternative and more efficient way using the saturation with respect to the homogeneous maximal ideal.

math.AC↗
arXiv2021

Private Information Retrieval Schemes Using Cyclic Codes

A Private Information Retrieval (PIR) scheme allows users to retrieve data from a database without disclosing to the server information about the identity of the data retrieved. A coded storage in a distributed storage system with colluding servers is considered in this work, namely the approach in [$t$-private information retrieval schemes using transitive codes, IEEE Trans. Inform. Theory, vol. 65, no. 4, pp. 2107-2118, 2019] which considers a storage and retrieval code with a transitive group and provides binary PIR schemes with the highest possible rate. Reed-Muller codes were considered in [$t$-private information retrieval schemes using transitive codes, IEEE Trans. Inform. Theory, vol. 65, no. 4, pp. 2107-2118, 2019]. In this work, we consider cyclic codes and we show that binary PIR schemes using cyclic codes provide a larger constellation of PIR parameters and they may outperform the ones coming from Reed-Muller codes in some cases.

cs.IT↗
arXiv2021

An Algorithmic Approach to Entanglement-Assisted Quantum Error-Correcting Codes from the Hermitian Curve

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 $c$, 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 $c$ for this family of EAQECCs. As a result, this algorithm allows us to provide EAQECCs with excellent parameters over any field size.

cs.IT↗
arXiv2020

Entanglement-Assisted Quantum Error Correcting Codes From RS Codes and BCH Codes with Extension Degree 2

Entanglement-assisted quantum error correcting codes (EAQECCs) constructed from Reed-Solomon codes and BCH codes are considered in this work. It is provided a complete and explicit formula for the parameters of EAQECCs coming from any Reed-Solomon code, for the Hermitian metric, and from any BCH code with extension degree $2$ and consecutive cyclotomic cosets, for both the Euclidean and the Hermitian metric. The main task in this work is the computation of a completely general formula for $c$, the minimum number of required maximally entangled quantum states.

cs.IT↗