Constructions and Properties of Linear Locally Repairable Codes (original) (raw)
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Optimal Locally Repairable Linear Codes
IEEE Journal on Selected Areas in Communications, 2014
Linear erasure codes with local repairability are desirable for distributed data storage systems. An [n, k, d] code having all-symbol (r, δ)-locality, denoted as (r, δ)a, is considered optimal if it also meets the minimum Hamming distance bound. The existing results on the existence and the construction of optimal (r, δ)a codes are limited to only the special case of δ = 2, and to only two small regions within this special case, namely, m = 0 or m ≥ (v +δ −1) > (δ −1), where m = n mod (r+δ −1) and v = k mod r. This paper investigates the existence conditions and presents deterministic constructive algorithms for optimal (r, δ)a codes with general r and δ. First, a structure theorem is derived for general optimal (r, δ)a codes which helps illuminate some of their structure properties. Next, the entire problem space with arbitrary n, k, r and δ is divided into eight different cases (regions) with regard to the specific relations of these parameters. For two cases, it is rigorously proved that no optimal (r, δ)a could exist. For four other cases the optimal (r, δ)a codes are shown to exist, deterministic constructions are proposed and the lower bound on the required field size for these algorithms to work is provided. Our new constructive algorithms not only cover more cases, but for the same cases where previous algorithms exist, the new constructions require a considerably smaller field, which translates to potentially lower computational complexity. Our findings substantially enriches the knowledge on (r, δ)a codes, leaving only two cases in which the existence of optimal codes are yet to be determined.
Explicit optimal-length locally repairable codes of distance 5
arXiv (Cornell University), 2018
Locally repairable codes (LRCs) have received significant recent attention as a method of designing data storage systems robust to server failure. Optimal LRCs offer the ideal trade-off between minimum distance and locality, a measure of the cost of repairing a single codeword symbol. For optimal LRCs with minimum distance greater than or equal to 5, block length is bounded by a polynomial function of alphabet size. In this paper, we give an explicit construction of optimal-length (in terms of alphabet size), optimal LRCs with minimum distance equal to 5. Erratum In an earlier version we presented a construction of explicit optimal-length locally repairable codes of distance 5 using cyclic codes, which however was incorrect and only had distance 4. For that reason the cyclic construction is omitted on this version.
Codes With Local Regeneration and Erasure Correction
IEEE Transactions on Information Theory, 2014
Regenerating codes and codes with locality are two coding schemes that have recently been proposed, which in addition to ensuring data collection and reliability, also enable efficient node repair. In a situation where one is attempting to repair a failed node, regenerating codes seek to minimize the amount of data downloaded for node repair, while codes with locality attempt to minimize the number of helper nodes accessed. This paper presents results in two directions. In one, this paper extends the notion of codes with locality so as to permit local recovery of an erased code symbol even in the presence of multiple erasures, by employing local codes having minimum distance >2. An upper bound on the minimum distance of such codes is presented and codes that are optimal with respect to this bound are constructed. The second direction seeks to build codes that combine the advantages of both codes with locality as well as regenerating codes. These codes, termed here as codes with local regeneration, are codes with locality over a vector alphabet, in which the local codes themselves are regenerating codes. We derive an upper bound on the minimum distance of vectoralphabet codes with locality for the case when their constituent local codes have a certain uniform rank accumulation property. This property is possessed by both minimum storage regeneration (MSR) and minimum bandwidth regeneration (MBR) codes. We provide several constructions of codes with local regeneration which achieve this bound, where the local codes are either MSR or MBR codes. Also included in this paper, is an upper bound on the minimum distance of a general vector code with locality as well as the performance comparison of various code constructions of fixed block length and minimum distance.
On the Combinatorics of Locally Repairable Codes via Matroid Theory
IEEE Transactions on Information Theory, 2016
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, δ) of LRCs are generalized to matroids, and the matroid analogue of the generalized Singleton bound in [P. Gopalan et al., "On the locality of codeword symbols," IEEE Trans. Inf. Theory] 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, δ) 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 in [W. Song et al., "Optimal locally repairable codes," IEEE J. Sel. Areas Comm.].
Codes with hierarchical locality
2015 IEEE International Symposium on Information Theory (ISIT), 2015
In this paper, we study the notion of codes with hierarchical locality that is identified as another approach to local recovery from multiple erasures. The well-known class of codes with locality is said to possess hierarchical locality with a single level. In a code with two-level hierarchical locality, every symbol is protected by an innermost local code, and another middle-level code of larger dimension containing the local code. We first consider codes with two levels of hierarchical locality, derive an upper bound on the minimum distance, and provide optimal code constructions of low field-size under certain parameter sets. Subsequently, we generalize both the bound and the constructions to hierarchical locality of arbitrary levels. Index Terms Codes with locality, locally recoverable codes, hierarchical locality, multiple erasures, distributed storage. I. INTRODUCTION An important desirable attribute in a distributed storage system is the efficiency in carrying out repair of failed nodes. Among many others, two important metrics to characterize efficiency of node repair are repair bandwidth, i.e., the amount of data download in the case of a node failure and repair degree, i.e., the number of helper nodes accessed for node repair. While regenerating codes [1] aim to minimize the repair bandwidth, codes with locality [2] seek to minimize the repair degree. The focus of the present paper is on codes with locality. A. Codes with Locality An [n, k, d] linear code C can possibly require to access k symbols to recover one lost symbol. The notion of locality of code symbols was introduced in [2], with the aim of designing codes in such a way that the number of symbols accessed to repair a lost symbol is much smaller than the dimension k of the code. The code C is said to have locality r if the i-th code symbol c i , 1 ≤ i ≤ n can be recovered by accessing r << k other code symbols. In [2], authors proved an upper bound on the minimum distance of codes with locality, and showed that an existing family of pyramid codes [3] can achieve the bound. In [4], authors extended the notion to (r, δ)-locality, where each symbol can be recovered locally even in the presence of an additional (δ − 2) erasures. In [2], authors introduced categories of information-symbol and all-symbol locality. In the former, local recoverability is guaranteed for symbols from an information set, while in the latter, it is guaranteed for every symbol. Explicit constructions for codes with all-symbol locality are provided in [5], [6], respectively based on rank-distance and Reed-Solomon (RS) codes. Improved bounds on the minimum distance of codes with all-symbol locality are provided in [7], [8], along with certain optimal constructions. Families of codes with all-symbol locality with small alphabet size (low field size) are constructed in [9]. Locally repairable codes over binary alphabet are constructed in [10]. A new approach of local regeneration, where in repair is both local and in addition bandwidth-efficient within the local group, achievable by making use of a vector alphabet is considered in [4], [11], [12]. Recently, many approaches are proposed in literature [4], [7], [9], [13], [14] to address the problem of recovering from multiple erasures locally. The notion of (r, δ)-locality introduced in [4] is one such. In [13], an approach of protecting a single symbol by multiple support-disjoint local codes of the same length is considered. An upper bound on the minimum distance is derived, and existence of optimal codes is established under certain constraints. A similar approach is considered in [9] also. In [9], authors allow multiple recovering sets of different sizes, and also provide constructions requiring field-size only in the order of block-length. Quite differently, authors of [7] consider codes allowing sequential recovery of two erasures, motivated by the fact that such a family of codes allow a larger minimum distance. An upper bound on the minimum distance and optimal constructions for restricted set of parameters are provided. B. Our Contributions In the present paper, we study the notion of hierarchical locality that is identified as another approach to local recovery from multiple erasures. In consideration of practical distributed storage systems, Duminuco et al. in [15] had proposed the topology
Cyclic Codes with Locality and Availability
ArXiv, 2018
In this work codes with availability are constructed based on the cyclic \emph{locally repairable code} (LRC) construction by Tamo et al. and their extension to (r,rho)(r,\rho)(r,rho)-locality by Chen et al. The minimum distance of these codes is increased by carefully extending their defining set. We give a bound on the dimension of LRCs with availability and orthogonal repair sets and show that the given construction is optimal for a range of parameters.
A family of codes with variable locality and availability
2021
In this work we present a class of locally recoverable codes, i.e. codes where an erasure at a position P of a codeword may be recovered from the knowledge of the entries in the positions of a recovery set RP . The codes in the class that we define have availability, meaning that for each position P there are several distinct recovery sets. Also, the entry at position P may be recovered even in the presence of erasures in some of the positions of the recovery sets, and the number of supported erasures may vary among the various recovery sets.
Binary cyclic codes that are locally repairable
2014 IEEE International Symposium on Information Theory, 2014
Codes for storage systems aim to minimize the repair locality, which is the number of disks (or nodes) that participate in the repair of a single failed disk. Simultaneously, the code must sustain a high rate, operate on a small finite field to be practically significant and be tolerant to a large number of erasures. To this end, we construct new families of binary linear codes that have an optimal dimension (rate) for a given minimum distance and locality. Specifically, we construct cyclic codes that are locally repairable for locality 2 and distances 2, 6 and 10. In doing so, we discover new upper bounds on the code dimension, and prove the optimality of enabling local repair by provisioning disjoint groups of disks. Finally, we extend our construction to build codes that have multiple repair sets for each disk.
2013 Information Theory and Applications Workshop (ITA), 2013
Regenerating codes and codes with locality are schemes recently proposed for a distributed storage network. While regenerating codes minimize data download for node repair, codes with locality minimize the number of nodes accessed during repair. In this paper, we provide constructions of codes with locality, in which the local codes are regenerating codes, thereby combining the advantages of both classes of codes. We also derive upper bounds on the minimum distance and code size for this class of codes and show that the proposed constructions achieve this bound. The constructions include both the cases when the local regenerating codes correspond to the MSR point as well as the MBR point on the storage repair-bandwidth tradeoff curve.
Codes with locality for two erasures
2014 IEEE International Symposium on Information Theory, 2014
In this paper, we study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. By a local parity-check computation, we mean recovery via a single parity-check equation associated to small Hamming weight. Earlier approaches considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. These codes, which we refer to as locally 2-reconstructible codes, are a natural generalization along one direction, of codes with all-symbol locality introduced by Gopalan et al, in which recovery from a single erasure is considered. By studying the Generalized Hamming Weights of the dual code, we derive upper bounds on the minimum distance of locally 2-reconstructible codes and provide constructions for a family of codes based on Turán graphs, that are optimal with respect to this bound. The minimum distance bound derived here is universal in the sense that no code which permits all-symbol local recovery from 2 erasures can have larger minimum distance regardless of approach adopted. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case.