Quasi-cyclic flexible regenerating codes are very interesting because of their simplicity and low... more Quasi-cyclic flexible regenerating codes are very interesting because of their simplicity and low complexity. They allow exact repair-by-transfer in the minimum bandwidth case and an exact pseudo repair-by-transfer in the MSR case, where operations are needed only when a new node enters into the system replacing a lost one.
... Using these con-structions, new families of quaternary ReedMuller (RM) codes are built with ... more ... Using these con-structions, new families of quaternary ReedMuller (RM) codes are built with the peculiarity that after using the Gray map the ob-tained 4-linear codes have the same parameters and fundamental properties as the codes in the usual binary linear RM family. ...
Storage optimization in distributed environments is a major concern when talking about reliabilit... more Storage optimization in distributed environments is a major concern when talking about reliability in this kind of schemes. Although replication is the most used option, erasure coding is a more optimized one. However, erasure coding uses a lot of bandwidth to replace one node. In a dynamic scheme, where nodes enter and leave the system frequently, bandwidth use could be an important drawback. Regenerating Codes introduced by Dimakis et al. minimize the code repair problem by applying Network Coding to the distributed storage scheme. However finding the coefficients for the linear combinations used to replace a node is not easy, specially for the systematic case, and must be calculated for each new node fail. For the special case when the dimension k and the redundancy n-k of the system are the same, and the number of nodes participating in the regeneration phase is d=k+1, we purpose a new optimal family of Minimum Storage Regenerating codes where coefficients are precalculated and ...
In this paper, the family of quasi-cyclic Minimum Storage Regen- erating codes is extended from t... more In this paper, the family of quasi-cyclic Minimum Storage Regen- erating codes is extended from the Minimum Storage Regenerating (MSR) to the Minimum Bandwidth Regenerating (MBR) codes. Quasi-cyclic MBR codes achieve exact and uncoded repair properties with a flexible parameter d. More- over, they achieve optimality in the MBR point of the tradeoff curve.
In a distributed storage environment, where the data is placed in nodes connected through a netwo... more In a distributed storage environment, where the data is placed in nodes connected through a network, it is likely that one of these nodes fails. It is known that the use of erasure coding improves the fault tolerance and minimizes the redundancy added in distributed storage environments. The use of regenerating codes not only make the most of the erasure coding improvements, but also minimizes the amount of data needed to regenerate a failed node. In this paper, a new family of regenerating codes based on quasi-cyclic codes is presented. Quasi-cyclic flexible minimum storage regenerating (QCFMSR) codes are constructed and their existence is proved. Quasi-cyclic flexible regenerating codes with minimum bandwidth constructed from a base QCFMSR code are also provided. These codes not only achieve optimal MBR parameters in terms of stored data and repair bandwidth, but also for an specific choice of the parameters involved, they can be decreased under the optimal MBR point. Quasi-cyclic...
In a realistic distributed storage environment, storage nodes are usually placed in racks, a meta... more In a realistic distributed storage environment, storage nodes are usually placed in racks, a metallic support designed to accommodate electronic equipment. It is known that the communication (bandwidth) cost between nodes which are in the same rack is much lower than between nodes which are in different racks. In this paper, a new model, where the storage nodes are placed in two racks, is proposed and analyzed. Moreover, the two-rack model is generalized to any number of racks. In this model, the storage nodes have different repair costs depending on the rack where they are placed. A threshold function, which minimizes the amount of stored data per node and the bandwidth needed to regenerate a failed node, is shown. This threshold function generalizes the ones given for previous distributed storage models. The tradeoff curve obtained from this threshold function is compared with the ones obtained from the previous models, and it is shown that this new model outperforms the previous ...
... Using these constructions new families of quaternary Reed-Muller codes are built with the pec... more ... Using these constructions new families of quaternary Reed-Muller codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the same parameters as the codes in the classical binary linear Reed-Muller family. ...
Abstract. Recently, new families of quaternary linear Reed-Muller codes such that, after the Gray... more Abstract. Recently, new families of quaternary linear Reed-Muller codes such that, after the Gray map, the corresponding Z4-linear codes have the same parameters and properties as the codes in the usual binary lin-ear Reed-Muller family have been introduced. A structural ...
New quaternary Plotkin constructions are given and are used to obtain new families of quaternary ... more New quaternary Plotkin constructions are given and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension and the minimum distance are studied. Using these constructions new families of quaternary Reed-Muller codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the
ABSTRACT A binary nonlinear code can be represented as a union of cosets of a binary linear subco... more ABSTRACT A binary nonlinear code can be represented as a union of cosets of a binary linear subcode. In this paper, the complexity of some algorithms to obtain this representation is analyzed. Moreover, some properties and constructions of new codes from given ones in terms of this representation are described. Algorithms to compute the minimum distance of binary nonlinear codes, based on known algorithms for linear codes, are also established, along with an algorithm to decode such codes. All results are written in such a way that they can be easily transformed into algorithms, and the performance of these algorithms is evaluated.
... Engineering Autonomous University of Barcelona Cerdanyola del Vallès (Barcelona), Spain {Bern... more ... Engineering Autonomous University of Barcelona Cerdanyola del Vallès (Barcelona), Spain {Bernat.Gaston | Jaume.Pujol | Merce.Villanueva}@uab.cat ... The use of network coding introduces some computational complexity [3], since it requires not only solving a system of ...
Quasi-cyclic flexible regenerating codes are very interesting because of their simplicity and low... more Quasi-cyclic flexible regenerating codes are very interesting because of their simplicity and low complexity. They allow exact repair-by-transfer in the minimum bandwidth case and an exact pseudo repair-by-transfer in the MSR case, where operations are needed only when a new node enters into the system replacing a lost one.
... Using these con-structions, new families of quaternary ReedMuller (RM) codes are built with ... more ... Using these con-structions, new families of quaternary ReedMuller (RM) codes are built with the peculiarity that after using the Gray map the ob-tained 4-linear codes have the same parameters and fundamental properties as the codes in the usual binary linear RM family. ...
Storage optimization in distributed environments is a major concern when talking about reliabilit... more Storage optimization in distributed environments is a major concern when talking about reliability in this kind of schemes. Although replication is the most used option, erasure coding is a more optimized one. However, erasure coding uses a lot of bandwidth to replace one node. In a dynamic scheme, where nodes enter and leave the system frequently, bandwidth use could be an important drawback. Regenerating Codes introduced by Dimakis et al. minimize the code repair problem by applying Network Coding to the distributed storage scheme. However finding the coefficients for the linear combinations used to replace a node is not easy, specially for the systematic case, and must be calculated for each new node fail. For the special case when the dimension k and the redundancy n-k of the system are the same, and the number of nodes participating in the regeneration phase is d=k+1, we purpose a new optimal family of Minimum Storage Regenerating codes where coefficients are precalculated and ...
In this paper, the family of quasi-cyclic Minimum Storage Regen- erating codes is extended from t... more In this paper, the family of quasi-cyclic Minimum Storage Regen- erating codes is extended from the Minimum Storage Regenerating (MSR) to the Minimum Bandwidth Regenerating (MBR) codes. Quasi-cyclic MBR codes achieve exact and uncoded repair properties with a flexible parameter d. More- over, they achieve optimality in the MBR point of the tradeoff curve.
In a distributed storage environment, where the data is placed in nodes connected through a netwo... more In a distributed storage environment, where the data is placed in nodes connected through a network, it is likely that one of these nodes fails. It is known that the use of erasure coding improves the fault tolerance and minimizes the redundancy added in distributed storage environments. The use of regenerating codes not only make the most of the erasure coding improvements, but also minimizes the amount of data needed to regenerate a failed node. In this paper, a new family of regenerating codes based on quasi-cyclic codes is presented. Quasi-cyclic flexible minimum storage regenerating (QCFMSR) codes are constructed and their existence is proved. Quasi-cyclic flexible regenerating codes with minimum bandwidth constructed from a base QCFMSR code are also provided. These codes not only achieve optimal MBR parameters in terms of stored data and repair bandwidth, but also for an specific choice of the parameters involved, they can be decreased under the optimal MBR point. Quasi-cyclic...
In a realistic distributed storage environment, storage nodes are usually placed in racks, a meta... more In a realistic distributed storage environment, storage nodes are usually placed in racks, a metallic support designed to accommodate electronic equipment. It is known that the communication (bandwidth) cost between nodes which are in the same rack is much lower than between nodes which are in different racks. In this paper, a new model, where the storage nodes are placed in two racks, is proposed and analyzed. Moreover, the two-rack model is generalized to any number of racks. In this model, the storage nodes have different repair costs depending on the rack where they are placed. A threshold function, which minimizes the amount of stored data per node and the bandwidth needed to regenerate a failed node, is shown. This threshold function generalizes the ones given for previous distributed storage models. The tradeoff curve obtained from this threshold function is compared with the ones obtained from the previous models, and it is shown that this new model outperforms the previous ...
... Using these constructions new families of quaternary Reed-Muller codes are built with the pec... more ... Using these constructions new families of quaternary Reed-Muller codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the same parameters as the codes in the classical binary linear Reed-Muller family. ...
Abstract. Recently, new families of quaternary linear Reed-Muller codes such that, after the Gray... more Abstract. Recently, new families of quaternary linear Reed-Muller codes such that, after the Gray map, the corresponding Z4-linear codes have the same parameters and properties as the codes in the usual binary lin-ear Reed-Muller family have been introduced. A structural ...
New quaternary Plotkin constructions are given and are used to obtain new families of quaternary ... more New quaternary Plotkin constructions are given and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension and the minimum distance are studied. Using these constructions new families of quaternary Reed-Muller codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the
ABSTRACT A binary nonlinear code can be represented as a union of cosets of a binary linear subco... more ABSTRACT A binary nonlinear code can be represented as a union of cosets of a binary linear subcode. In this paper, the complexity of some algorithms to obtain this representation is analyzed. Moreover, some properties and constructions of new codes from given ones in terms of this representation are described. Algorithms to compute the minimum distance of binary nonlinear codes, based on known algorithms for linear codes, are also established, along with an algorithm to decode such codes. All results are written in such a way that they can be easily transformed into algorithms, and the performance of these algorithms is evaluated.
... Engineering Autonomous University of Barcelona Cerdanyola del Vallès (Barcelona), Spain {Bern... more ... Engineering Autonomous University of Barcelona Cerdanyola del Vallès (Barcelona), Spain {Bernat.Gaston | Jaume.Pujol | Merce.Villanueva}@uab.cat ... The use of network coding introduces some computational complexity [3], since it requires not only solving a system of ...
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