Optimal linear codes with a local-error-correction property (original) (raw)

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.

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.

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

Constructions and Properties of Linear Locally Repairable Codes

IEEE Transactions on Information Theory, 2016

In this paper, locally repairable codes with all-symbol locality are studied. Methods to modify already existing codes are presented. Also, it is shown that with high probability, a random matrix with a few extra columns guaranteeing the locality property, is a generator matrix for a locally repairable code with a good minimum distance. The proof of this also gives a constructive method to find locally repairable codes. Constructions are given of three infinite classes of optimal vector-linear locally repairable codes over an alphabet of small size, not depending on the size of the code.

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.

On one generalization of LRC codes with availability

2017 IEEE Information Theory Workshop (ITW)

We investigate one possible generalization of locally recoverable codes (LRC) with all-symbol locality and availability when recovering sets can intersect in a small number of coordinates. This feature allows us to increase the achievable code rate and still meet load balancing requirements. In this paper we derive an upper bound for the rate of such codes and give explicit constructions of codes with such a property. These constructions utilize LRC codes developed by Wang et al.

Erasure codes with simplex locality

We focus on erasure codes for distributed storage. The distributed storage setting imposes locality requirements because of easy repair demands on the decoder. We first establish the characterization of various locality properties in terms of the generator matrix of the code. These lead to bounds on locality and notions of optimality. We then examine the locality properties of a family of nonbinary codes with simplex structure. We investigate their optimality and design several easy repair decoding methods. In particular, we show that any correctable erasure pattern can be solved by easy repair.

A Family of Codes With Locality Containing Optimal Codes

IEEE Access

Locally recoverable codes were introduced by Gopalan et al. in 2012, and in the same year Prakash et al. introduced the concept of codes with locality, which are a type of locally recoverable codes. In this work we introduce a new family of codes with locality, which are subcodes of a certain family of evaluation codes. We determine the dimension of these codes, and also bounds for the minimum distance. We present the true values of the minimum distance in special cases, and also show that some elements of this family are ''optimal codes'', as defined by Prakash et al. INDEX TERMS Locally recoverable codes, affine cartesian codes. II. A FAMILY OF LOCALLY RECOVERABLE CODES Let F q be a finite field with q elements. Definition 2.1: Let m, r, δ be positive integers, with δ ≥ 2 and r +δ−1 ≤ m. We say that a (linear) code C ⊂ F m q is (r, δ)locally recoverable if for every i ∈ {1,. .. , m} there exists a subset S i ⊂ {1,. .. , m}, containing i and of cardinality at most r + δ − 1, such that the punctured code obtained by removing the entries which are not in S i has minimum distance at least δ. The condition on the minimum distance in the above definition shows that one cannot have two distinct codewords in the punctured code which coincide in (at least) r positions,

Linear programming bounds for distributed storage codes

Advances in Mathematics of Communications

A major issue of locally repairable codes is their robustness. If a local repair group is not able to perform the repair process, this will result in increasing the repair cost. Therefore, it is critical for a locally repairable code to have multiple repair groups. In this paper we consider robust locally repairable coding schemes which guarantee that there exist multiple distinct (not necessarily disjoint) alternative local repair groups for any single failure such that the failed node can still be repaired locally even if some of the repair groups are not available. We use linear programming techniques to establish upper bounds on the code size of these codes. We also provide two examples of robust locally repairable codes that are optimal regarding our linear programming bound. Furthermore, we address the update efficiency problem of the distributed data storage networks. Any modification on the stored data will result in updating the content of the storage nodes. Therefore, it is essential to minimise the number of nodes which need to be updated by any change in the stored data. We characterise the update-efficient storage code properties and establish the necessary conditions of existence update-efficient locally repairable storage codes.

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.