arXiv subjects

Rubén Rodríguez-Ballesteros

Publications and source records attributed to Rubén Rodríguez-Ballesteros.

3 recordsLinked to original sources

arXiv2026

On the structure of polynomial ideal codes and additive Reed--Solomon codes

We study structural properties of polynomial ideal (PI) codes and additive Reed--Solomon (aRS) codes in the folded Hamming metric (equivalently, the Hamming metric over a larger alphabet). We first determine the maximum possible length of PI codes, prove that the dual of every PI code is a generalized PI code, and deduce that PI codes are dually QMDS together with an explicit formula for their weight distribution. We then show that PI codes defined by irreducible polynomials correspond to aRS codes with pairwise nonconjugate evaluation points. For these aRS codes, the previous results on PI codes apply (duality, QMDS property and weight distribution). We show that, asymptotically, aRS codes with nonconjugate evaluation points achieve the same length as PI codes. We also compute the permutation group of such aRS codes. We also extend the family of PI codes to consider arbitrary polynomial modules (not necessarily pairwise coprime), and study when these and general aRS codes satisfy the QMDS and dually QMDS properties.

cs.IT↗

arXiv2026

Upper bounds on the length of quasi-MDS codes

We study upper bounds on the length of $\mathbb F_q$-linear QMDS codes in the folded Hamming distance relative to their other parameters, especially the field size $q$. Via a correspondence between such codes and families of subspaces, we relate the length problem to that of upper bounding $1$-subspace packings with respect to the other parameters, especially the field size. Our main result is a reduction from these families to partial spreads, which allows us to import sharp bounds from finite geometry, including results of Drake-Freeman, Năstase-Sissokho, and Honold-Kiermaier-Kurz. As a consequence, we recover the Griesmer-type upper bound on the length of QMDS codes by Ball et al. and obtain tighter upper bounds in several parameter regimes.

cs.IT↗

arXiv2024

Linear codes in the folded Hamming distance and the quasi MDS property

In this work, we study linear codes with the folded Hamming distance, or equivalently, codes with the classical Hamming distance that are linear over a subfield. This includes additive codes. We study MDS codes in this setting and define quasi MDS (QMDS) codes and dually QMDS codes, which attain a more relaxed variant of the classical Singleton bound. We provide several general results concerning these codes, including restriction, shortening, weight distributions, existence, density, geometric description and bounds on their lengths relative to their field sizes. We provide explicit examples and a binary construction with optimal lengths relative to their field sizes, which beats any MDS code.

cs.IT↗