arXiv subjects

Adrián Fidalgo-Díaz

Publications and source records attributed to Adrián Fidalgo-Díaz.

4 recordsLinked to original sources

arXiv2025

The Clifford defect of a numerical semigroup

The Clifford defect is a rational number associated to the Weierstrass semigroup at a given point of an algebraic curve. It describes the error-correcting capability of the so-called Modified Algorithm for decoding the corresponding one-point codes defined at the point. This defect also finds applications in other contexts involving one-point codes. We study the Clifford defect of some numerical semigroups arising from curves and give explicit formulas for them.

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 Gröbner 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↗

arXiv2024

Distributed matrix multiplication with straggler tolerance over very small field

The problem of distributed matrix multiplication with straggler tolerance over finite fields is considered, focusing on field sizes for which previous solutions were not applicable (for instance, the field of two elements). We employ Reed-Muller-type codes for explicitly constructing the desired algorithms and study their parameters by translating the problem into a combinatorial problem involving sums of discrete convex sets. We generalize polynomial codes and matdot codes, discussing the impossibility of the latter being applicable for very small field sizes, while providing optimal solutions for some regimes of parameters in both cases.

cs.IT↗

arXiv2024

Distributed matrix multiplication with straggler tolerance using algebraic function fields

The problem of straggler mitigation in distributed matrix multiplication (DMM) is considered for a large number of worker nodes and a fixed small finite field. Polynomial codes and matdot codes are generalized by making use of algebraic function fields (i.e., algebraic functions over an algebraic curve) over a finite field. The construction of optimal solutions is translated to a combinatorial problem on the Weierstrass semigroups of the corresponding algebraic curves. Optimal or almost optimal solutions are provided. These have the same computational complexity per worker as classical polynomial and matdot codes, and their recovery thresholds are almost optimal in the asymptotic regime (growing number of workers and a fixed finite field).

cs.IT↗