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Optimal ternary locally repairable codes

DESIGNS CODES AND CRYPTOGRAPHY 2024年第9期92卷 2685-2704页

作者： Hao, Jie Xia, Shu-Tao Shum, Kenneth W. Chen, Bin Fu, Fang-Wei Yang, Yixian Beijing Univ Posts & Telecommun Natl Engn Res Ctr Disaster Backup & Recovery Beijing 100876 Peoples R China Tsinghua Univ Tsinghua Shenzhen Int Grad Sch Shenzhen 518055 Peoples R China Chinese Univ Hong Kong Shenzhen Sch Sci & Engn Shenzhen 518172 Peoples R China Harbin Inst Technol Dept Comp Sci & Technol Shenzhen 518055 Peoples R China Nankai Univ Chern Inst Math Tianjin 300071 Peoples R China Nankai Univ LPMC Tianjin 300071 Peoples R China Peng Cheng Lab Res Ctr Artificial Intelligence Shenzhen 518055 Peoples R China

Locally repairable codes (LRCs) are linear codes with locality properties for code symbols, which have important applications in distributed storage systems. In this paper, we completely classify all the possible code parameters of optimal ternary LRCs achieving the Singleton-like bound proposed by Gopalan et al. Explicit constructions of optimal ternary LRCs are given for each group of possible code parameters. Moreover, it is also proved that optimal ternary LRCs with maximal minimum distance 6 are unique up to the equivalence of linear codes.

关键词： Locally repairable codes coding for distributed storage systems Locality Singleton-like bound

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Anticode-based locally repairable codes with high availability

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DESIGNS CODES AND CRYPTOGRAPHY 2018年第2期86卷 419-445页

作者： Silberstein, Natalia Zeh, Alexander Technion Israel Inst Technol Dept Comp Sci Haifa Israel Ben Gurion Univ Negev Dept Elect & Comp Engn Beer Sheva Israel Yahoo Labs Haifa Israel Infineon Technol AG Neubiberg Germany

This paper presents constructions of new families of locally repairable codes (LRCs) with small locality and high availability, where each code symbol can be recovered by using many (exponential in the dimension of the code) disjoint small sets (of size 2 or 3) of other code symbols. Following the method of Farrell, the generator matrices of our LRCs are obtained by deleting certain columns from the generator matrix of the Simplex code, where the deleted columns form different anticodes. Most of the resulting codes, defined over any finite field and in particular over the binary field, are optimal either with respect to the Griesmer bound, or with respect to the Cadambe-Mazumdar bound for LRCs, or both.

关键词： Anticodes Availability coding for distributed storage systems Locally repairable codes

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Locality and Availability in distributed storage

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IEEE TRANSACTIONS ON INFORMATION THEORY 2016年第8期62卷 4481-4493页

作者： Rawat, Ankit Singh Papailiopoulos, Dimitris S. Dimakis, Alexandros G. Vishwanath, Sriram Univ Texas Austin Dept Elect & Comp Engn Austin TX 78712 USA Carnegie Mellon Univ Dept Comp Sci Pittsburgh PA 15213 USA Univ Calif Berkeley Dept Elect Engn & Comp Sci Berkeley CA 94720 USA Univ Texas Austin Dept Elect Engn & Comp Sci Austin TX 78712 USA

This paper studies the problem of information symbol availability in codes: we refer to a systematic code as code with (r, t)-availability if every information (systematic) symbol can be reconstructed from t disjoint groups of other code symbols, each of the sizes at most r. This paper shows that it is possible to construct codes that can support a scaling number of parallel reads while keeping the rate to be an arbitrarily high constant. It further shows that this is possible with the minimum Hamming distance arbitrarily close to the Singleton bound. This paper also presents a bound demonstrating a tradeoff between rate, minimum Hamming distance, and availability parameters. Our codes match the aforementioned bound, and their constructions rely on certain combinatorial structures. Resolvable designs provide one way to realize these required combinatorial structures. The two constructions presented in this paper require field sizes, which are linear and exponential in the code length, respectively. From a practical standpoint, our codes are relevant for distributed storage applications involving hot data, i.e., the information, which is frequently accessed by multiple processes in parallel.

关键词： coding for distributed storage systems locally repairable codes codes with availability rank-metric codes

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On the Combinatorics of Locally Repairable Codes via Matroid Theory

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IEEE TRANSACTIONS ON INFORMATION THEORY 2016年第10期62卷 5296-5315页

作者： Westerback, Thomas Freij-Hollanti, Ragnar Ernvall, Toni Hollanti, Camilla Aalto Univ Dept Math & Syst Anal POB 11100 FI-00076 Aalto Finland Aalto Univ Dept Commun & Networking POB 13000 FI-00076 Aalto Finland Aalto Univ Dept Math & Syst Anal FI-00076 Espoo Finland Univ Turku Dept Math & Stat FI-20014 Turku Finland

This paper provides a link between matroid theory and locally repairable codes (LRCs) that are either linear or more generally almost affine. Using this link, new results on both LRCs and matroid theory are derived. The parameters (n, k, d, r, delta) of LRCs are generalized to matroids, and the matroid analog of the generalized singleton bound by Gopalan et al. for linear LRCs is given for matroids. It is shown that the given bound is not tight for certain classes of parameters, implying a nonexistence result for the corresponding locally repairable almost affine codes that are coined perfect in this paper. Constructions of classes of matroids with a large span of the parameters (n, k, d, r, delta) and the corresponding local repair sets are given. Using these matroid constructions, new LRCs are constructed with prescribed parameters. The existence results on linear LRCs and the nonexistence results on almost affine LRCs given in this paper strengthen the nonexistence and existence results on perfect linear LRCs given by Song et al.

关键词： coding for distributed storage systems locally repairable codes linear codes almost affine codes matroids

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On the Combinatorics of Locally Repairable Codes via Matroid Theory

On the Combinatorics of Locally Repairable Codes via Matroid...

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IEEE Information Theory Workshop (ITW)

关键词： coding for distributed storage systems locally repairable codes linear codes almost affine codes matroids

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Error-Correcting Regenerating and Locally Repairable Codes via Rank-Metric Codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2015年第11期61卷 5765-5778页

作者： Silberstein, Natalia Rawat, Ankit Singh Vishwanath, Sriram Technion Israel Inst Technol Dept Comp Sci IL-32000 Haifa Israel Univ Texas Austin Dept Elect & Comp Engn Austin TX 78712 USA

This paper presents and analyzes a novel concatenated coding scheme for enabling error resilience in two distributed storage settings: one being storage using existing regenerating codes and the second being storage using locally repairable codes. The concatenated coding scheme brings together a maximum rank distance code as an outer code and either a globally regenerating or a locally repairable code as an inner code. In addition, error resilience for combination of locally repairable codes with regenerating codes is considered. This concatenated coding system is designed to handle two different types of adversarial errors: the first type includes an adversary that can replace the content of an affected node only once;while the second type studies an adversary that is capable of polluting data an unbounded number of times. The paper establishes an upper bound on the resilience capacity for a locally repairable code. This paper also proves that the proposed concatenated coding approach attains the upper bound on the resilience capacity in the presence of the first type of adversary for both minimum storage regenerating codes and locally repairable codes. Further, this paper presents mechanisms that combine the presented concatenated coding scheme with subspace signatures to achieve error resilience for the second type of errors.

关键词： coding for distributed storage systems error-correcting codes rank-metric codes regenerating codes locally repairable codes

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Optimal Fractional Repetition Codes Based on Graphs and Designs

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IEEE TRANSACTIONS ON INFORMATION THEORY 2015年第8期61卷 4164-4180页

作者： Silberstein, Natalia Etzion, Tuvi Technion Israel Inst Technol Dept Comp Sci IL-32000 Haifa Israel

Fractional repetition (FR) codes is a family of codes for distributed storage systems (DSSs) that allow for uncoded exact repairs having the minimum repair bandwidth. However, in contrast to minimum bandwidth regenerating (MBR) codes, where an arbitrary set of a certain size of available nodes is used for a node repair, the repairs with FR codes are table based. This usually allows to store more data compared with MBR codes. In this paper, we consider bounds on the FR capacity, which is the maximum amount of data that can be stored using an FR code. Optimal FR codes which attain these bounds are presented. The constructions of these FR codes are based on combinatorial designs and on families of regular and biregular graphs. These constructions of FR codes for given parameters raise some interesting questions in graph theory. These questions and some of their solutions are discussed in this paper. In addition, based on a connection between FR codes and batch codes, we propose a new family of codes for DSS, namely, FR batch codes, which have the properties of batch codes and FR codes simultaneously. These are the first codes for DSS which allow for uncoded efficient exact repairs and load balancing which can be performed by several users in parallel. Other concepts related to FR codes are also discussed.

关键词： coding for distributed storage systems fractional repetition codes combinatorial batch codes Turan graphs cages transversal designs generalized polygons

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Optimal Locally Repairable and Secure Codes for distributed storage systems

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IEEE TRANSACTIONS ON INFORMATION THEORY 2014年第1期60卷 212-236页

作者： Rawat, Ankit Singh Koyluoglu, Onur Ozan Silberstein, Natalia Vishwanath, Sriram Univ Texas Austin Dept Elect & Comp Engn Austin TX 78712 USA

This paper aims to go beyond resilience into the study of security and local-repairability for distributed storage systems (DSSs). Security and local-repairability are both important as features of an efficient storage system, and this paper aims to understand the trade-offs between resilience, security, and local-repairability in these systems. In particular, this paper first investigates security in the presence of colluding eavesdroppers, where eavesdroppers are assumed to work together in decoding the stored information. Second, this paper focuses on coding schemes that enable optimal local repairs. It further brings these two concepts together to develop locally repairable coding schemes for DSS that are secure against eavesdroppers. The main results of this paper include: 1) an improved bound on the secrecy capacity for minimum storage regenerating codes;2) secure coding schemes that achieve the bound for some special cases;3) a new bound on minimum distance for locally repairable codes;4) code construction for locally repairable codes that attain the minimum distance bound;and 5) repair-bandwidth-efficient locally repairable codes with and without security constraints.

关键词： coding for distributed storage systems locally repairable codes repair bandwidth efficient codes security

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Secure Cooperative Regenerating Codes for distributed storage systems

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IEEE TRANSACTIONS ON INFORMATION THEORY 2014年第9期60卷 5228-5244页

作者： Koyluoglu, Onur Ozan Rawat, Ankit Singh Vishwanath, Sriram Univ Arizona Dept Elect & Comp Engn Tucson AZ 85721 USA Univ Texas Austin Dept Elect & Comp Engn Austin TX 78712 USA

Regenerating codes enable trading off repair bandwidth for storage in distributed storage systems (DSS). Due to their distributed nature, these systems are intrinsically susceptible to attacks, and they may also be subject to multiple simultaneous node failures. Cooperative regenerating codes allow bandwidth efficient repair of multiple simultaneous node failures. This paper analyzes storage systems that employ cooperative regenerating codes that are robust to (passive) eavesdroppers. The analysis is divided into two parts, studying both minimum bandwidth and minimum storage cooperative regenerating scenarios. First, the secrecy capacity for minimum bandwidth cooperative regenerating codes is characterized. Second, for minimum storage cooperative regenerating codes, a secure file size upper bound and achievability results are provided. These results establish the secrecy capacity for the minimum storage scenario for certain special cases. In all scenarios, the achievability results correspond to exact repair, and secure file size upper bounds are obtained using min-cut analyses over a suitable secrecy graph representation of DSS. The main achievability argument is based on an appropriate precoding of the data to eliminate the information leakage to the eavesdropper.

关键词： coding for distributed storage systems cooperative repair minimum bandwidth cooperative regenerating (MBCR) codes minimum storage cooperative regenerating (MSCR) codes security

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Secure Locally Repairable Codes for distributed storage systems

Secure Locally Repairable Codes for Distributed Storage Syst...

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IEEE International Symposium on Information Theory (ISIT)

作者： Rawat, Ankit Singh Koyluoglu, O. Ozan Silberstein, Natalia Vishwanath, Sriram Univ Texas Austin Dept ECE Austin TX 78751 USA

ISBN: (纸本)9781479904464

This paper presents coding schemes for distributed storage systems (DSS) that are secure against eavesdroppers, while simultaneously enabling efficient node repair (regeneration). Towards this, novel upper bounds on secrecy capacity for minimum storage regenerating (MSR) codes and locally repairable codes (LRCs) are derived. The eavesdropper model considered in this paper incorporates the ability to listen in on data downloaded during l(2) node repairs in addition to content stored on l(1) nodes. Finally, this paper presents coding schemes, based on precoding using Gabidulin codes, that achieve the upper bounds on secrecy capacity and characterize the secrecy capacity of DSS for various settings of system parameters.

关键词： coding for distributed storage systems locally repairable codes minimum storage regenerating codes security

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