arXiv subjects

Delio Jaramillo-Velez

Publications and source records attributed to Delio Jaramillo-Velez.

8 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↗

arXiv2026

Private neighbors, perfect codes and their relation with the $\mathtt{v}$-number of closed neighborhood ideals

In this work, we investigate the connections between dominating sets, private neighbors, and perfect codes in graphs, and their relationships with commutative algebra. In particular, we estimate the $\mathtt{v}$-number of closed neighborhood ideals in terms of minimal dominating sets and private neighbors. We show how the $\mathtt{v}$-number is related to other graph invariants, such as the cover number, domination number, and matching number. Moreover, we explore the relation with the Castelnuovo-Mumford regularity, proving that the $\mathtt{v}$-number is a lower bound for the regularity of bipartite, very well-covered, and chordal graphs. Finally, drawing from the relation between efficient dominating set and perfect codes, we use the redundancy of Hamming codes to present lower and upper bounds for the $\mathtt{v}$-number of some special family of graphs.

math.AC↗

arXiv2025

Perfectly-Private Analog Secure Aggregation in Federated Learning

In federated learning, multiple parties train models locally and share their parameters with a central server, which aggregates them to update a global model. To address the risk of exposing sensitive data through local models, secure aggregation via secure multiparty computation has been proposed to enhance privacy. At the same time, perfect privacy can only be achieved by a uniform distribution of the masked local models to be aggregated. This raises a problem when working with real valued data, as there is no measure on the reals that is invariant under the masking operation, and hence information leakage is bound to occur. Shifting the data to a finite field circumvents this problem, but as a downside runs into an inherent accuracy complexity tradeoff issue due to fixed point modular arithmetic as opposed to floating point numbers that can simultaneously handle numbers of varying magnitudes. In this paper, a novel secure parameter aggregation method is proposed that employs the torus rather than a finite field. This approach guarantees perfect privacy for each party's data by utilizing the uniform distribution on the torus, while avoiding accuracy losses. Experimental results show that the new protocol performs similarly to the model without secure aggregation while maintaining perfect privacy. Compared to the finite field secure aggregation, the torus-based protocol can in some cases significantly outperform it in terms of model accuracy and cosine similarity, hence making it a safer choice.

cs.LG↗

arXiv2024

The edge code of hypergraphs

Given a hypergraph $\mathcal{H}$, we introduce a new class of evaluation toric codes called edge codes derived from $\mathcal{H}$. We analyze these codes, focusing on determining their basic parameters. We provide estimations for the minimum distance, particularly in scenarios involving $d$-uniform clutters. Additionally, we demonstrate that these codes exhibit self-orthogonality. Furthermore, we compute the minimum distances of edge codes for all graphs with five vertices.

math.AC↗

arXiv2023

Connected domination in graphs and v-numbers of binomial edge ideals

The v-number of a graded ideal is an algebraic invariant introduced by Cooper et al., and originally motivated by problems in algebraic coding theory. In this paper we study the case of binomial edge ideals and we establish a significant connection between their v-numbers and the concept of connected domination in graphs. More specifically, we prove that the localization of the v-number at one of the minimal primes of the binomial edge ideal $J_G$ of a graph $G$ coincides with the connected domination number of the defining graph, providing a first algebraic description of the connected domination number. As an immediate corollary, we obtain a sharp combinatorial upper bound for the v-number of binomial edge ideals of graphs. Lastly, building on some known results on edge ideals, we analyze how the v-number of $J_G$ behaves under Gröbner degeneration when $G$ is a closed graph.

math.AC↗

arXiv2021

Relative generalized Hamming weights of evaluation codes

The aim of this work is to algebraically describe the relative generalized Hamming weights of evaluation codes. We give a lower bound for these weights in terms of a footprint bound. We prove that this bound can be sharp. We compute the next-to-minimal weight of toric codes over hypersimplices of degree 1.

cs.IT↗

arXiv2020

Coding theory package for Macaulay2

In this Macaulay2 \cite{M2} package we define an object called {\it linear code}. We implement functions that compute basic parameters and objects associated with a linear code, such as generator and parity check matrices, the dual code, length, dimension, and minimum distance, among others. We define an object {\it evaluation code}, a construction which allows to study linear codes using tools of algebraic geometry and commutative algebra. We implement functions to generate important families of linear codes such as Hamming codes, cyclic codes, Reed--Solomon codes, Reed--Muller codes, Cartesian codes, monomial--Cartesian codes, and toric codes. In addition, we define functions for the syndrome decoding algorithm and locally recoverable code construction, which are important tools in applications of linear codes. The package \textit{CodingTheory.m2} is available at \url{https://github.com/Macaulay2/Workshop-2020-Cleveland/tree/CodingTheory/CodingTheory}

cs.IT↗

arXiv2020

$F$-thresholds and test ideals of Thom-Sebastiani type polynomials

We provide a formula for $F$-thresholds of a Thom-Sebastiani type polynomial over a perfect field of prime characteristic. This result extends the formula for the $F$-pure threshold of a diagonal hypersurface. We also compute the first test ideal of Thom-Sebastiani type polynomials. Finally, we apply our result to find hypersurfaces where the log canonical thresholds equals the $F$-pure thresholds for infinitely many prime numbers.

math.AC↗