Non-homogeneous Two-Rack Model for Distributed Storage Systems

In the traditional two-rack distributed storage system (DSS) model, due to the assumption that the storage capacity of each node is the same, the minimum bandwidth regenerating (MBR) point becomes infeasible. In this paper, we design a new non-homogeneous two-rack model by proposing a generalization of the threshold function used to compute the tradeoff curve. We prove that by having the nodes in the rack with higher regenerating bandwidth stores more information, all the points on the tradeoff curve, including the MBR point, become feasible. Finally, we show how the non-homogeneous two-rack model outperforms the traditional model in the tradeoff curve between the storage per node and the repair bandwidth.Comment: ISIT 2013. arXiv admin note: text overlap with arXiv:1004.0785 by other author

Case 170: Pericardial fat necrosis

History: A 30-year-old man presented to our emergency department with acute pleuritic chest pain. He had no fever, dyspnea, or other symptoms, nor did he have a history of chest trauma. A physical examination yielded normal findings. Laboratory test results and electrocardiographic findings were normal. Axial chest computed tomography (CT) was performed

Codes over Z4 and permutation decoding of linear codes

Aquest treball es va publicar com "Linear Codes over the Integer Residue Ring Z4" in Handbook of magma funcions / edited by John Cannon, Wieb Bosma, Claus Fieker and Allan Steel (2017), p. 5575-5616

On Quaternary linear Reed-Muller codes

A la literatura recent hi podem trobar la introducció de noves famílies de codis de Reed- Muller quaternaris lineals RMs. Les imatges d'aquests nous codis a través del mapa de Gray són codis binaris Z4-lineals que comparteixen els paràmetres i les propietats (longitud, dimensió, distància mínima, inclusió, i relació de dualitat) amb la família de codis de Reed- Muller binaris lineals. El kernel d'un codi binari C es defineix com K(C) = {x 2 Zn2 : C + x = C}. La dimensió del kernel és un invariant estructural per els codis binaris equivalents. Part d'aquesta tesi consisteix en establir els valors de la dimensió del kernel per aquestes noves famílies de codis de Reed-Muller Z4-lineals. Tot i que dos codis Z4- lineals no equivalents poden compartir el mateix valor de la dimensió del kernel, en el cas dels codis de Reed-Muller RMs aquest resultat es suficient per donar-ne una classificació completa. Per altra banda, un codi quaternari lineal de Hadamard C, és un codi que un cop li hem aplicat el mapa de Gray obtenim un codi binari de Hadamard. És conegut que els codis de Hadamard quaternaris formen part de les famílies de codis quaternaris de Reed- MullerRMs. Definim el grup de permutacions d'un codi quaternari lineal com PAut(C) = { 2 Sn : (C) = C}. Com a resultat d'aquesta tesi també s'estableix l'ordre dels grups de permutacions de les famílies de codis de Hadamard quaternaris. A més a més, aquests grups són caracteritzats proporcionant la forma dels seus generadors i la forma de les òrbites del grup PAut(C) actuant sobre el codi C. Sabem que el codi dual, en el sentit quaternari, d'un codi de Hadamard és un codi 1-perfecte estès. D'aquesta manera els resultats obtinguts sobre el grup de permutacions es poden transportar a una família de codis quaternaris 1- perfectes estesosRecently, new families of quaternary linear Reed-Muller codes RMs have been introduced. They satisfy that, under the Gray map, the corresponding Z4-linear codes have the same parameters and properties (length, dimension, minimum distance, inclusion, and duality relation) as the codes of the binary linear Reed-Muller family. The kernel of a binary code C is K(C) = {x 2 Zn2 : C + x = C}. The dimension of the kernel is a structural invariant for equivalent binary codes. In this work, the dimension of the kernel for these new families of Z4-linear Reed-Muller codes is established. This result is sufficient to give a full classification of these new families of Z4-linear Reed-Muller codes up to equivalence. A quaternary linear Hadamard code C is a code over Z4 that under the Gray map, the corresponding Z4-linear code is a binary Hadamard code. It is well known that quaternary linear Hadamard codes are included in the RMs families of codes. The permutation automorphism group of a quaternary linear code C of length n is defined as PAut(C) = { 2 Sn : (C) = C}. In this dissertation, the order of the permutation automorphism group of all quaternary linear Hadamard codes is established. Moreover, these groups are completely characterized by providing their generators and also by computing the orbits of their action on C. Since the dual of a Hadamard code is an extended 1-perfect code in the quaternary sense, the permutation automorphism group of the quaternary linear extended 1-perfect codes is also established

On Quaternary Linear Reed-Muller Codes

A la literatura recent hi podem trobar la introducció de noves famílies de codis de Reed- Muller quaternaris lineals RMs. Les imatges d’aquests nous codis a través del mapa de Gray són codis binaris Z4-lineals que comparteixen els paràmetres i les propietats (longitud, dimensió, distància mínima, inclusió, i relació de dualitat) amb la família de codis de Reed- Muller binaris lineals. El kernel d’un codi binari C es defineix com K(C) = {x 2 Zn2 : C + x = C}. La dimensió del kernel és un invariant estructural per els codis binaris equivalents. Part d’aquesta tesi consisteix en establir els valors de la dimensió del kernel per aquestes noves famílies de codis de Reed-Muller Z4-lineals. Tot i que dos codis Z4- lineals no equivalents poden compartir el mateix valor de la dimensió del kernel, en el cas dels codis de Reed-Muller RMs aquest resultat es suficient per donar-ne una classificació completa. Per altra banda, un codi quaternari lineal de Hadamard C, és un codi que un cop li hem aplicat el mapa de Gray obtenim un codi binari de Hadamard. És conegut que els codis de Hadamard quaternaris formen part de les famílies de codis quaternaris de Reed- MullerRMs. Definim el grup de permutacions d’un codi quaternari lineal com PAut(C) = {  2 Sn :  (C) = C}. Com a resultat d’aquesta tesi també s’estableix l’ordre dels grups de permutacions de les famílies de codis de Hadamard quaternaris. A més a més, aquests grups són caracteritzats proporcionant la forma dels seus generadors i la forma de les òrbites del grup PAut(C) actuant sobre el codi C. Sabem que el codi dual, en el sentit quaternari, d’un codi de Hadamard és un codi 1-perfecte estès. D’aquesta manera els resultats obtinguts sobre el grup de permutacions es poden transportar a una família de codis quaternaris 1- perfectes estesosRecently, new families of quaternary linear Reed-Muller codes RMs have been introduced. They satisfy that, under the Gray map, the corresponding Z4-linear codes have the same parameters and properties (length, dimension, minimum distance, inclusion, and duality relation) as the codes of the binary linear Reed-Muller family. The kernel of a binary code C is K(C) = {x 2 Zn2 : C + x = C}. The dimension of the kernel is a structural invariant for equivalent binary codes. In this work, the dimension of the kernel for these new families of Z4-linear Reed-Muller codes is established. This result is sufficient to give a full classification of these new families of Z4-linear Reed-Muller codes up to equivalence. A quaternary linear Hadamard code C is a code over Z4 that under the Gray map, the corresponding Z4-linear code is a binary Hadamard code. It is well known that quaternary linear Hadamard codes are included in the RMs families of codes. The permutation automorphism group of a quaternary linear code C of length n is defined as PAut(C) = {  2 Sn :  (C) = C}. In this dissertation, the order of the permutation automorphism group of all quaternary linear Hadamard codes is established. Moreover, these groups are completely characterized by providing their generators and also by computing the orbits of their action on C. Since the dual of a Hadamard code is an extended 1-perfect code in the quaternary sense, the permutation automorphism group of the quaternary linear extended 1-perfect codes is also established

SYSTEM AND METHODS FOR DISTRIBUTED DATA STORAGE

A systematic distributed storage system (DSS) comprising: a plurality of storage nodes, wherein each storage node configures to store a plurality of sub blocks of a data file and a plurality of coded blocks, a set of repair pairs for each of the storage nodes, wherein the system is configured to use the respective repair pair of storage nodes to repair a lost or damaged sub block or coded block on a given storage node. Also a distributed storage system DSS comprising h non-empty nodes, and data stored non homogenously across the non-empty nodes according to the storing codes (n,k). Further a method for determining linear erasure codes with local repairability comprising: selecting two or more coding parameters including r and δ; determining if an optimal [n, k, d] code having all-symbol (r, δ)-locality (“(r, δ)a”) exists for the selected r, δ; and if the optimal (r, δ)a code exists performing a local repairable code using the optimal (r, δ)a code