arXiv subjects

Umberto Martínez-Peñas

Publications and source records attributed to Umberto Martínez-Peñas.

At least 19 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.

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arXiv2026

Integer sequences that are generalized weights of a linear code

Which integer sequences are sequences of generalized weights of a linear code? In this paper, we answer this question for linear block codes, rank-metric codes, and more generally for sum-rank metric codes. We do so under an existence assumption for MDS and MSRD codes. We also prove that the same integer sequences appear as sequences of greedy weights of linear block codes, rank-metric codes, and sum-rank metric codes. Finally, we characterize the integer sequences which appear as sequences of relative generalized weights (respectively, relative greedy weights) of linear block codes.

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arXiv2026

Maximally recoverable codes with locality and availability

In this work, we introduce maximally recoverable codes with locality and availability. We consider locally repairable codes (LRCs) where certain subsets of $ t $ symbols belong each to $ N $ local repair sets, which are pairwise disjoint after removing the $ t $ symbols, and which are of size $ r+δ-1 $ and can correct $ δ-1 $ erasures locally. Classical LRCs with $ N $ disjoint repair sets and LRCs with $ N $-availability are recovered when setting $ t = 1 $ and $ t=δ-1=1 $, respectively. Allowing $ t > 1 $ enables our codes to reduce the storage overhead for the same locality and availability. In this setting, we define maximally recoverable LRCs (MR-LRCs) as those that can correct any globally correctable erasure pattern given the locality and availability constraints. We then identify a large class of global erasure patterns that can be corrected by such MR-LRCs and prove that they are all the correctable patterns when $ t=1 $. We provide three explicit constructions of LRCs that can correct such erasure patterns (thus MR-LRCs for $ t=1 $), based on MSRD codes, each attaining the smallest finite-field sizes for some parameter regime. Finally, we extend the known lower bound on finite-field sizes from classical MR-LRCs to our setting (for any value of $ t $).

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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.

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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.

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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.

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arXiv2024

Private Information Retrieval from Locally Repairable Databases with Colluding Servers

We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage codes (LRCs). We consider any number of local groups $ g $, locality $ r $, local distance $ δ$ and dimension $ k $. Our main contribution is a PIR scheme for maximally recoverable (MR) LRCs based on linearized Reed--Solomon codes, which achieve the smallest field sizes among MR-LRCs for many parameter regimes. In our scheme, nodes are identified with codeword symbols and servers are identified with local groups of nodes. Only locally non-redundant information is downloaded from each server, that is, only $ r $ nodes (out of $ r+δ-1 $) are downloaded per server. The PIR scheme achieves the (download) rate $ R = (N - k - rt + 1)/N $, where $ N = gr $ is the length of the MDS code obtained after removing the local parities, and for any $ t $ colluding servers such that $ k + rt \leq N $. For an unbounded number of stored files, the obtained rate is strictly larger than those of known PIR schemes that work for any MDS code. Finally, the obtained PIR scheme can also be adapted when communication between the user and each server is performed via linear network coding, achieving the same rate as previous PIR schemes for this scenario but with polynomial finite field sizes, instead of exponential. Our rates are equal to those of PIR schemes for Reed--Solomon codes, but Reed--Solomon codes are incompatible with the MR-LRC property or linear network coding, thus our PIR scheme is less restrictive in its applications.

cs.IT↗

arXiv2024

New constructions of MSRD codes

In this work, we provide four methods for constructing new maximum sum-rank distance (MSRD) codes. The first method, a variant of cartesian products, allows faster decoding than known MSRD codes of the same parameters. The other three methods allow us to extend or modify existing MSRD codes in order to obtain new explicit MSRD codes for sets of matrix sizes (numbers of rows and columns in different blocks) that were not attainable by previous constructions. In this way, we show that MSRD codes exist (by giving explicit constructions) for new ranges of parameters, in particular with different numbers of rows and columns at different positions.

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↗

arXiv2024

Maximum Sum-Rank Distance Codes over Finite Chain Rings

In this work, maximum sum-rank distance (MSRD) codes and linearized Reed-Solomon codes are extended to finite chain rings. It is proven that linearized Reed-Solomon codes are MSRD over finite chain rings, extending the known result for finite fields. For the proof, several results on the roots of skew polynomials are extended to finite chain rings. These include the existence and uniqueness of minimum-degree annihilator skew polynomials and Lagrange interpolator skew polynomials. A general cubic-complexity sum-rank Welch-Berlekamp decoder and a quadratic-complexity sum-rank syndrome decoder (under some assumptions) are then provided over finite chain rings. The latter also constitutes the first known syndrome decoder for linearized Reed--Solomon codes over finite fields. Finally, applications in Space-Time Coding with multiple fading blocks and physical-layer multishot Network Coding are discussed.

cs.IT↗

arXiv2023

Sum-rank metric codes

Sum-rank metric codes are a natural extension of both linear block codes and rank-metric codes. They have several applications in information theory, including multishot network coding and distributed storage systems. The aim of this chapter is to present the mathematical theory of sum-rank metric codes, paying special attention to the $\mathbb{F}_q$-linear case in which different sizes of matrices are allowed. We provide a comprehensive overview of the main results in the area. In particular, we discuss invariants, optimal anticodes, and MSRD codes. In the last section, we concentrate on $\mathbb{F}_{q^m}$-linear codes.

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arXiv2023

Zipper Codes

Zipper codes are a framework for describing spatially-coupled product-like codes. Many well-known codes, such as staircase codes and braided block codes, are subsumed into this framework. New types of codes such as tiled diagonal and delayed diagonal zipper codes are introduced along with their software simulation results. Stall patterns that can arise in iterative decoding are analyzed, giving a means of error floor estimation.

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arXiv2022

Doubly and triply extended MSRD codes

In this work, doubly extended linearized Reed--Solomon codes and triply extended Reed--Solomon codes are generalized. We obtain a general result in which we characterize when a multiply extended code for a general metric attains the Singleton bound. We then use this result to obtain several families of doubly extended and triply extended maximum sum-rank distance (MSRD) codes that include doubly extended linearized Reed--Solomon codes and triply extended Reed--Solomon codes as particular cases. To conclude, we discuss when these codes are one-weight codes.

cs.IT↗

arXiv2022

Zeros with multiplicity, Hasse derivatives and linear factors of general skew polynomials

In this work, multiplicities of zeros of skew polynomials are studied. Two distinct definitions are considered: First, $ a $ is said to be a zero of $ F $ of multiplicity $ r $ if $ (x-a)^r $ divides $ F $ on the right; second, $ a $ is said to be a zero of $ F $ of multiplicity $ r $ if some skew polynomial $ P = (x-a_r) \cdots (x-a_2) (x-a_1) $, having $ a_1 = a $ as its only right zero, divides $ F $ on the right. Neither of these two notions implies the other for general skew polynomials. We show that, in the first case, Lam and Leroy's concept of P-independence does not behave naturally, whereas a union theorem still holds. In contrast, we show that P-independence behaves naturally for the second notion of multiplicities. As a consequence, we provide extensions of classical commutative results to general skew polynomials. These include: (1) The upper bound on the number of (P-independent) zeros (counting multiplicities) of a skew polynomial by its degree, and (2) The equivalence of P-independence, Hermite interpolation and the invertibility of confluent Vandermonde matrices (for which we introduce skew polynomial Hasse derivatives).

math.RA↗

arXiv2022

A general family of MSRD codes and PMDS codes with smaller field sizes from extended Moore matrices

We construct six new explicit families of linear maximum sum-rank distance (MSRD) codes, each of which has the smallest field sizes among all known MSRD codes for some parameter regime. Using them and a previous result of the author, we provide two new explicit families of linear partial MDS (PMDS) codes with smaller field sizes than previous PMDS codes for some parameter regimes. Our approach is to characterize evaluation points that turn extended Moore matrices into the parity-check matrix of a linear MSRD code. We then produce such sequences from codes with good Hamming-metric parameters. The six new families of linear MSRD codes with smaller field sizes are obtained using MDS codes, Hamming codes, BCH codes and three Algebraic-Geometry codes. The MSRD codes based on Hamming codes, of minimum sum-rank distance $ 3 $, meet a recent bound by Byrne et al.

cs.IT↗

arXiv2022

Multilayer crisscross error and erasure correction

In this work, multilayer crisscross error and erasures are considered, which affect entire rows and columns in the matrices of a list of matrices. To measure such errors and erasures, the multi-cover metric is introduced. Several bounds are derived, including a Singleton bound, and maximum multi-cover distance (MMCD) codes are defined as those attaining it. Duality, puncturing and shortening of linear MMCD codes are studied. It is shown that the dual of a linear MMCD code is not necessarily MMCD, and those satisfying this duality condition are defined as dually MMCD codes. Finally, some constructions of codes in the multi-cover metric are given, including dually MMCD codes, together with efficient decoding algorithms for them.

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