DSpace Comunidad:http://hdl.handle.net/10045/339642020-09-26T16:30:48Z2020-09-26T16:30:48ZLocally Repairable Convolutional Codes With Sliding Window RepairMartínez-Peñas, UmbertoNapp, Diegohttp://hdl.handle.net/10045/1084112020-07-31T00:13:09Z2020-08-01T00:00:00ZTítulo: Locally Repairable Convolutional Codes With Sliding Window Repair
Autor/es: Martínez-Peñas, Umberto; Napp, Diego
Resumen: 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 generally, ∂−1 erasures per local group), and sliding-window global repair, which can correct erasure patterns with up to dcj−1 erasures in every window of j+1 consecutive blocks of n nodes, where dcj−1 is the j th column distance of the code. The parameter j can be adjusted, for a fixed LRCC, according to different catastrophic erasure patterns, requiring only to contact n(j+1)−dcj+1 nodes, plus less than μn other nodes, in the storage system, where μ is the memory of the code. A Singleton-type bound is provided for dcj−1 . If it attains such a bound, an LRCC can correct the same number of catastrophic erasures in a window of length n(j+1) as an optimal locally repairable block code of the same rate and locality, and with block length n(j+1) . In addition, the LRCC is able to perform the flexible and somehow local sliding-window repair by adjusting j . Furthermore, by adjusting and/or sliding the window, the LRCC can potentially correct more erasures in the original window of n(j+1) nodes than an optimal locally repairable block code of the same rate and locality, and length n(j+1) . Finally, the concept of partial maximum distance profile (partial MDP) codes is introduced. Partial MDP codes can correct all information-theoretically correctable erasure patterns for a given locality, local distance and information rate. An explicit construction of partial MDP codes whose column distances attain the provided Singleton-type bound, up to certain parameter j=L , is obtained based on known maximum sum-rank distance convolutional codes.2020-08-01T00:00:00ZA construction of F2-linear cyclic, MDS codesCardell, Sara D.Climent, Joan-JosepPanario, DanielStevens, Bretthttp://hdl.handle.net/10045/1080692020-07-16T00:11:04Z2020-08-01T00:00:00ZTítulo: A construction of F2-linear cyclic, MDS codes
Autor/es: Cardell, Sara D.; Climent, Joan-Josep; Panario, Daniel; Stevens, Brett
Resumen: In this paper we construct F2-linear codes over Fb2 with length n and dimension n−r where n=rb. These codes have good properties, namely cyclicity, low density parity-check matrices and maximum distance separation in some cases. For the construction, we consider an odd prime p, let n=p−1 and utilize a partition of Zn. Then we apply a Zech logarithm to the elements of these sets and use the results to construct an index array which represents the parity-check matrix of the code. These codes are always cyclic and the density of the parity-check and the generator matrices decreases to 0 as n grows (for a fixed r). When r=2 we prove that these codes are always maximum distance separable. For higher r some of them retain this property.2020-08-01T00:00:00ZRepresentations of Generalized Self-Shrunken SequencesCardell, Sara D.Climent, Joan-JosepFúster Sabater, AmparoRequena Arévalo, Verónicahttp://hdl.handle.net/10045/1076132020-06-26T00:06:40Z2020-06-19T00:00:00ZTítulo: Representations of Generalized Self-Shrunken Sequences
Autor/es: Cardell, Sara D.; Climent, Joan-Josep; Fúster Sabater, Amparo; Requena Arévalo, Verónica
Resumen: Output sequences of the cryptographic pseudo-random number generator, known as the generalized self-shrinking generator, are obtained self-decimating Pseudo-Noise (PN)-sequences with shifted versions of themselves. In this paper, we present three different representations of this family of sequences. Two of them, the p and G-representations, are based on the parameters p and G corresponding to shifts and binary vectors, respectively, used to compute the shifted versions of the original PN-sequence. In addition, such sequences can be also computed as the binary sum of diagonals of the Sierpinski’s triangle. This is called the B-representation. Characteristics and generalities of the three representations are analyzed in detail. Under such representations, we determine some properties of these cryptographic sequences. Furthermore, these sequences form a family that has a group structure with the bit-wise XOR operation.2020-06-19T00:00:00ZOn formations of monoidsBranco, Mário J.J.Gomes, Gracinda M.S.Pin, Jean-ÉricSoler-Escrivà, Xarohttp://hdl.handle.net/10045/1074332020-06-18T00:10:15Z2020-11-01T00:00:00ZTítulo: On formations of monoids
Autor/es: Branco, Mário J.J.; Gomes, Gracinda M.S.; Pin, Jean-Éric; Soler-Escrivà, Xaro
Resumen: A formation of monoids is a class of finite monoids closed under taking quotients and subdirect products. Formations of monoids were first studied in connection with formal language theory, but in this paper, we come back to an algebraic point of view. We give two natural constructions of formations based on constraints on the minimal ideal and on the maximal subgroups of a monoid. Next we describe two sublattices of the lattice of all formations, and give, for each of them, an isomorphism with a known lattice of varieties of monoids. Finally, we study formations and varieties containing only Clifford monoids, completely describe such varieties and discuss the case of formations.2020-11-01T00:00:00Z