63 citations · 117 across the 14 of their papers we have counts for
21 papers
Optimal anticodes, MSRD codes, and generalized weights in the sum-rank metric
Eduardo Camps Moreno, Elisa Gorla, Cristina Landolina +3
Sum-rank metric codes have recently attracted the attention of many researchers, due to their relevance in several applications. Mathematically, the sum-rank metric is a natural ge…
Systematic Maximum Sum Rank Codes
Paulo Almeida, Umberto Martínez-Penas, Diego Napp
In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This…
Hamming and simplex codes for the sum-rank metric
Umberto Martínez-Peñas
Sum-rank Hamming codes are introduced in this work. They are essentially defined as the longest codes (thus of highest information rate) with minimum sum-rank distance at least $ 3…
Private Information Retrieval from Locally Repairable Databases with Colluding Servers
Umberto Martínez-Peñas
We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage code…
Locally Repairable Convolutional Codes with Sliding Window Repair
Umberto Martínez-Peñas, Diego Napp
Locally repairable convolutional codes (LRCCs) for distributed storage systems (DSSs) are introduced in this work. They enable local repair, for a single node erasure (or more gene…
Universal and Dynamic Locally Repairable Codes with Maximal Recoverability via Sum-Rank Codes
Umberto Martínez-Peñas, Frank R. Kschischang
Locally repairable codes (LRCs) are considered with equal or unequal localities, local distances and local field sizes. An explicit two-layer architecture with a sum-rank outer cod…
Theory of supports for linear codes endowed with the sum-rank metric
Umberto Martínez-Peñas
The sum-rank metric naturally extends both the Hamming and rank metrics in coding theory over fields. It measures the error-correcting capability of codes in multishot matrix-multi…
Reliable and Secure Multishot Network Coding using Linearized Reed-Solomon Codes
Umberto Martínez-Peñas, Frank R. Kschischang
Multishot network coding is considered in a worst-case adversarial setting in which an omniscient adversary with unbounded computational resources may inject erroneous packets in u…
Skew and linearized Reed-Solomon codes and maximum sum rank distance codes over any division ring
Umberto Martínez-Peñas
Reed-Solomon codes and Gabidulin codes have maximum Hamming distance and maximum rank distance, respectively. A general construction using skew polynomials, called skew Reed-Solomo…
Bounding the number of common zeros of multivariate polynomials and their consecutive derivatives
Olav Geil, Umberto Martínez-Peñas
We upper bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-…
Universal secure rank-metric coding schemes with optimal communication overheads
Umberto Martínez-Peñas
We study the problem of reducing the communication overhead from a noisy wire-tap channel or storage system where data is encoded as a matrix, when more columns (or their linear co…
Relative generalized matrix weights of matrix codes for universal security on wire-tap networks
Umberto Martínez-Peñas, Ryutaroh Matsumoto
Universal security over a network with linear network coding has been intensively studied. However, previous linear codes and code pairs used for this purpose were linear over a la…
Communication efficient and strongly secure secret sharing schemes based on algebraic geometry codes
Umberto Martínez-Peñas
Secret sharing schemes with optimal and universal communication overheads have been obtained independently by Bitar et al. and Huang et al. However, their constructions require a f…
Unifying notions of generalized weights for universal security on wire-tap networks
Umberto Martínez-Peñas, Ryutaroh Matsumoto
Universal security over a network with linear network coding has been intensively studied. However, previous linear codes used for this purpose were linear over a larger field than…
Generalized rank weights of reducible codes, optimal cases and related properties
Umberto Martínez-Peñas
Reducible codes for the rank metric were introduced for cryptographic purposes. They have fast encoding and decoding algorithms, include maximum rank distance (MRD) codes and can c…
Rank equivalent and rank degenerate skew cyclic codes
Umberto Martínez-Peñas
Two skew cyclic codes can be equivalent for the Hamming metric only if they have the same length, and only the zero code is degenerate. The situation is completely different for th…
Rank error-correcting pairs
Umberto Martínez-Peñas, Ruud Pellikaan
Error-correcting pairs were introduced independently by Pellikaan and Kötter as a general method of decoding linear codes with respect to the Hamming metric using coordinatewise pr…
Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth's second tower
Olav Geil, Stefano Martin, Umberto Martínez-Peñas +1
Asymptotically good sequences of ramp secret sharing schemes were given in [Asymptotically good ramp secret sharing schemes, arXiv:1502.05507] by using one-point algebraic geometri…
On the roots and minimum rank distance of skew cyclic codes
Umberto Martínez-Peñas
Skew cyclic codes play the same role as cyclic codes in the theory of error-correcting codes for the rank metric. In this paper, we give descriptions of these codes by root spaces,…
On the similarities between generalized rank and Hamming weights and their applications to network coding
Umberto Martínez-Peñas
Rank weights and generalized rank weights have been proven to characterize error and erasure correction, and information leakage in linear network coding, in the same way as Hammin…
On asymptotically good ramp secret sharing schemes
Olav Geil, Stefano Martin, Umberto Martínez-Peñas +2
Asymptotically good sequences of linear ramp secret sharing schemes have been intensively studied by Cramer et al. in terms of sequences of pairs of nested algebraic geometric code…