Rate Optimal Binary Linear Locally Repairable Codes with Small Availability
Swanand Kadhe, Robert Calderbank
备注：
Shorter version of the paper was presented at ISIT 2017. The updated second version contains a new section on rate upper bounds for binary linear codes with (2,t) and (r,3)-availability
A locally repairable code with availability has the property that every code symbol can be recovered from multiple, disjoint subsets of other symbols of small size. In particular, a code symbol is said to have $(r,t)$-availability if it can be recovered from $t$ disjoint subsets, each of size at most $r$. A code with availability is said to be 'rate-optimal', if its rate is maximum among the class of codes with given locality, availability, and alphabet size.
This paper focuses on rate-optimal binary, linear codes with small availability, and makes four contributions. First, it establishes tight upper bounds on the rate of binary linear codes with $(r,2)$ and $(2,3)$ availability. Second, it establishes a uniqueness result for binary rate-optimal codes, showing that for certain classes of binary linear codes with $(r,2)$ and $(2,3)$-availability, any rate optimal code must be a direct sum of shorter rate optimal codes. Third, it presents novel upper bounds on the rates of binary linear codes with $(2,t)$ and $(r,3)$-availability. In particular, the main contribution here is a new method for bounding the number of cosets of the dual of a code with availability, using its covering properties. Finally, it presents a class of locally repairable linear codes associated with convex polyhedra, focusing on the codes associated with the Platonic solids. It demonstrates that these codes are locally repairable with $t = 2$, and that the codes associated with (geometric) dual polyhedra are (coding theoretic) duals of each other.
Locally repairable codes are desirable for distributed storage systems to improve the repair efficiency. In this paper, we first build a bridge between locally repairable code and packing. As an application of this bridge, some optimal locally repairable codes can be obtained by packings, which gives optimal locally repairable codes with flexible parameters.
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 gives also a constructive method to find locally repairable codes.
Locally repairable codes have been investigated extensively in recent years due to practical applications in distributed and cloud storage systems. However, there are few asymptotical constructions of locally repairable codes in the literature. In this paper, we provide an explicit asymptotic construction of locally repairable codes over arbitrary finite fields from local expansions of functions at a rational place. This construction gives a Tsfasman-Vladut-Zink type bound for locally repairable codes. Its main advantage is that there are no constraints on both locality and alphabet size. Furt…
In classical coding theory, it is common to construct new codes via propagation rules. There are various propagation rules to construct classical block codes. However, propagation rules have not been extensively explored for constructions of locally repairable codes. In this paper, we introduce a few propagation rules to construct good locally repairable codes. To our surprise, these simple propagation rules produce a few interesting results. Firstly, by concatenating a locally repairable code as an inner code with a classical block code as an outer code, we obtain quite a few dimension-optima…
In this work, we give locally repairable regenerating code (LRRC) [1]-[3], [5], [6] constructions that can protect the file size promised by the graph analysis of the modified family helper selection (MFHS) scheme [1] at the minimum-bandwidth-regenerating (MBR) point.
Typically, locally repairable codes (LRCs) and regenerating codes have been studied independently of each other, and it has not been clear how the parameters of one relate to those of the other. In this paper, a novel connection between locally repairable codes and exact regenerating codes is established. Via this connection, locally repairable codes are interpreted as exact regenerating codes. Further, some of these codes are shown to perform better than time-sharing codes between minimum bandwidth regenerating and minimum storage regenerating codes.
Locally repairable codes have become a key instrument in large-scale distributed storage systems. This paper focuses on the construction of locally repairable codes with $(r,δ)$-locality that achieve the equality in the Singleton-type bound. We use matrix-product codes to propose two infinite families of $q$-ary optimal $(r,δ)$ locally repairable codes of lengths up to $q^2+q$. The ingredients in the matrix-product codes are either linear maximum distance separable codes or optimal locally repairable codes of small lengths. Further analysis and refinement yield a construction of another infini…
This paper presents a construction for several families of optimal binary locally repairable codes (LRCs) with small locality (2 and 3). This construction is based on various anticodes. It provides binary LRCs which attain the Cadambe-Mazumdar bound. Moreover, most of these codes are optimal with respect to the Griesmer bound.
Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code {\it optimal} if it achieves the Singleton-type bound). In the breakthrough work of \cite{TB14}, several classes of optimal locally repairable codes were constructed via subcodes of Reed-Solomon codes. Thus, the lengths of the codes given in \cite{TB14} are upper bounded by the code alphabet size $q$. Recently, it was proved through extension of construction in \cite{TB14} that length of $q$-ary optimal locally repairable codes can be $q+1$ in \cite{JMX17}. Surprisingly,…
Constructing locally repairable codes achieving Singleton-type bound (we call them optimal codes in this paper) is a challenging task and has attracted great attention in the last few years. Tamo and Barg \cite{TB14} first gave a breakthrough result in this topic by cleverly considering subcodes of Reed-Solomon codes. Thus, $q$-ary optimal locally repairable codes from subcodes of Reed-Solomon codes given in \cite{TB14} have length upper bounded by $q$. Recently, it was shown through extension of construction in \cite{TB14} that length of $q$-ary optimal locally repairable codes can be $q+1$ i…
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.
In this survey, two related families of codes are discussed: batch codes and codes for private information retrieval. These two families can be viewed as natural generalizations of locally repairable codes, which were extensively studied in the context of coding for fault tolerance in distributed data storage systems. Bounds on the parameters of the codes, as well as basic constructions, are presented. Connections between different code families are discussed.
Locally repairable codes which are optimal with respect to the bound presented by Prakash et al. are considered. New upper bounds on the length of such optimal codes are derived. The new bounds both improve and generalize previously known bounds. Optimal codes are constructed, whose length is order-optimal when compared with the new upper bounds. The length of the codes is super-linear in the alphabet size.
We consider error decoding of locally repairable codes (LRC) and partial MDS (PMDS) codes through interleaved decoding. For a specific class of LRCs we investigate the success probability of interleaved decoding. For PMDS codes we show that there is a wide range of parameters for which interleaved decoding can increase their decoding radius beyond the minimum distance with the probability of successful decoding approaching $1$, when the code length goes to infinity.
New asymptotic upper bounds are presented on the rate of sequences of locally repairable codes (LRCs) with a prescribed relative minimum distance and locality over a finite field $F$. The bounds apply to LRCs in which the recovery functions are linear; in particular, the bounds apply to linear LRCs over $F$. The new bounds are shown to improve on previously published results, especially when the repair groups are disjoint, namely, they form a partition of the set of coordinates.
Locally repairable codes enables fast repair of node failure in a distributed storage system. The code symbols in a codeword are stored in different storage nodes, such that a disk failure can be recovered by accessing a small fraction of the storage nodes. The number of storage nodes that are contacted during the repair of a failed node is a parameter called locality. We consider locally repairable codes that can be locally recovered in the presence of multiple node failures. The punctured code obtained by removing the code symbols in the complement of a repair group is called a local code. W…
Locally repairable codes, or locally recoverable codes (LRC for short) are designed for application in distributed and cloud storage systems. Similar to classical block codes, there is an important bound called the Singleton-type bound for locally repairable codes. In this paper, an optimal locally repairable code refers to a block code achieving this Singleton-type bound. Like classical MDS codes, optimal locally repairable codes carry some very nice combinatorial structures. Since introduction of the Singleton-type bound for locally repairable codes, people have put tremendous effort on cons…
We show that locally repairable codes (LRCs) can be list decoded efficiently beyond the Johnson radius for a large range of parameters by utilizing the local error correction capabilities. The new decoding radius is derived and the asymptotic behavior is analyzed. We give a general list decoding algorithm for LRCs that achieves this radius along with an explicit realization for a class of LRCs based on Reed-Solomon codes (Tamo-Barg LRCs). Further, a probabilistic algorithm for unique decoding of low complexity is given and its success probability analyzed.
This paper aims to go beyond resilience into the study of security and local-repairability for distributed storage systems (DSS). 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 stored information. Second, the paper focuses on coding schemes that enable optimal local repa…
This paper presents a new explicit construction for locally repairable codes (LRCs) for distributed storage systems which possess all-symbols locality and maximal possible minimum distance, or equivalently, can tolerate the maximal number of node failures. This construction, based on maximum rank distance (MRD) Gabidulin codes, provides new optimal vector and scalar LRCs. In addition, the paper also discusses mechanisms by which codes obtained using this construction can be used to construct LRCs with efficient repair of failed nodes by combination of LRC with regenerating codes.
We study the Singleton-type bound that provides an upper limit on the minimum distance of locally repairable codes. We present an improved bound by carefully analyzing the combinatorial structure of the repair sets. Thus, we show the previous bound is unachievable for certain parameters. We then also provide explicit constructions of optimal codes that show that for certain parameters the new bound is sharp. Additionally, as a byproduct, some previously known codes are shown to attain the new bound and are thus proved to be optimal.
Locally repairable codes (LRC) have recently been a subject of intense research due to theoretical appeal and their application in distributed storage systems. In an LRC, any coordinate of a codeword can be recovered by accessing only few other coordinates. For LRCs over small alphabet (such as binary), the optimal rate-distance trade-off is unknown. In this paper we provide the tightest known upper bound on the rate of linear LRCs of a given relative distance, an improvement over any previous result, in particular \cite{cadambe2013upper}.
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 (MRD) code as an outer code and either a globally regenerating or a locally repairable code as an inner code. Also, error resilience for combination of locally repairable codes with regenerating codes is considered. This concatenated coding system is designed to handle two differe…
A locally repairable code is called Singleton-optimal if it achieves the Singleton-type bound. Such codes are of great theoretic interest in the study of locally repairable codes. In the recent years there has been a great amount of work on this topic. One of the main problems in this topic is to determine the largest length of a q-ary Singleton-optimal locally repairable code for given locality and minimum distance. Unlike classical MDS codes, the maximum length of Singleton? Optimal locally repairable codes are very sensitive to minimum distance and locality. Thus, it is more challenging and…
Locally repairable codes have been investigated extensively in recent years due to practical application in distributed storage as well as theoretical interest. However, not much work on asymptotical behavior of locally repairable codes has been done until now. In particular, there is a little result on constructive lower bound on asymptotical behavior of locally repairable codes. In this paper, we extend the construction given in \cite{BTV17} via automorphism groups of function field towers. The main advantage of our construction is to allow more flexibility of locality. Furthermore, we show …
For a systematic erasure code, update complexity (UC) is defined as the maximum number of parity blocks needed to be changed when some information blocks are updated. Locally repairable codes (LRCs) have been recently proposed and used in real-world distributed storage systems. In this paper, update complexity for optimal LRC is studied and both lower and upper bounds on UC are established in terms of length (n), dimension (k), minimum distance (d), and locality (r) of the code, when (r+1)|n. Furthermore, a class of optimal LRCs with small UC is proposed. Our proposed LRCs could be of interest…
Constructions of optimal locally repairable codes (LRCs) in the case of $(r+1) \nmid n$ and over small finite fields were stated as open problems for LRCs in [I. Tamo \emph{et al.}, "Optimal locally repairable codes and connections to matroid theory", \emph{2013 IEEE ISIT}]. In this paper, these problems are studied by constructing almost optimal linear LRCs, which are proven to be optimal for certain parameters, including cases for which $(r+1) \nmid n$. More precisely, linear codes for given length, dimension, and all-symbol locality are constructed with almost optimal minimum distance. `Alm…
Locally repairable codes(LRCs) play important roles in distributed storage systems(DSS). LRCs with small locality have their own advantages since fewer available symbols are needed in the recovery of erased symbols. In this paper, we prove an upper bound on the dimension of LRCs with minimum distance $d\geq 7$. An upper bound on the length of almost optimal LRCs with $d=7$, $r=2$ at $q^2+q+3$ is proved. Then based on the $t$-spread structure, we give an algorithm to construct almost optimal LRCs with $d=7$, $r=2$ and length $n\geq 3\lceil\frac{\sqrt{2}q}{3}\rceil$ when $q\geq 4$, whose dimensi…
In this paper we extend the notion of {\em locally repairable} codes to {\em secret sharing} schemes. The main problem that we consider is to find optimal ways to distribute shares of a secret among a set of storage-nodes (participants) such that the content of each node (share) can be recovered by using contents of only few other nodes, and at the same time the secret can be reconstructed by only some allowable subsets of nodes. As a special case, an eavesdropper observing some set of specific nodes (such as less than certain number of nodes) does not get any information. In other words, we p…
In an $[n,k,d]$ linear code, a code symbol is said to have locality $r$ if it can be repaired by accessing at most $r$ other code symbols. For an $(n,k,r)$ \emph{locally repairable code} (LRC), the minimum distance satisfies the well-known Singleton-like bound $d\le n-k-\lceil k/r\rceil +2$. In this paper, we study optimal ternary LRCs meeting this Singleton-like bound by employing a parity-check matrix approach. It is proved that there are only $8$ classes of possible parameters with which optimal ternary LRCs exist. Moreover, we obtain explicit constructions of optimal ternary LRCs for all t…
For binary $[n,k,d]$ linear locally repairable codes (LRCs), two new upper bounds on $k$ are derived. The first one applies to LRCs with disjoint local repair groups, for general values of $n,d$ and locality $r$, containing some previously known bounds as special cases. The second one is based on solving an optimization problem and applies to LRCs with arbitrary structure of local repair groups. Particularly, an explicit bound is derived from the second bound when $d\geq 5$. A specific comparison shows this explicit bound outperforms the Cadambe-Mazumdar bound for $5\leq d\leq 8$ and large val…
Locally repairable codes (LRC) for distribute storage allow two approaches to locally repair multiple failed nodes: 1) parallel approach, by which each newcomer access a set of $r$ live nodes $(r$ is the repair locality$)$ to download data and recover the lost packet; and 2) sequential approach, by which the newcomers are properly ordered and each newcomer access a set of $r$ other nodes, which can be either a live node or a newcomer ordered before it. An $[n,k]$ linear code with locality $r$ and allows local repair for up to $t$ failed nodes by sequential approach is called an $(n,k,r,t)$-exa…
In this paper, we propose locally repairable codes (LRCs) with optimal minimum distance for distributed storage systems (DSS). A two-layer encoding structure is employed to ensure data reconstruction and the designated repair locality. The data is first encoded in the first layer by any existing maximum distance separable (MDS) codes, and then the encoded symbols are divided into non-overlapping groups and encoded by an MDS array code in the second layer. The encoding in the second layer provides enough redundancy for local repair, while the overall code performs recovery of the data based on …
Optimal locally repairable codes with information locality are considered. Optimal codes are constructed, whose length is also order-optimal with respect to a new bound on the code length derived in this paper. The length of the constructed codes is super-linear in the alphabet size, which improves upon the well known pyramid codes, whose length is only linear in the alphabet size. The recoverable erasure patterns are also analyzed for the new codes. Based on the recoverable erasure patterns, we construct generalized sector-disk (GSD) codes, which can recover from disk erasures mixed with sect…
Constructions of optimal locally repairable codes (LRCs) achieving Singleton-type bound have been exhaustively investigated in recent years. In this paper, we consider new bounds and constructions of Singleton-optimal LRCs with minmum distance $d=6$, locality $r=3$ and minimum distance $d=7$ and locality $r=2$, respectively. Firstly, we establish equivalent connections between the existence of these two families of LRCs and the existence of some subsets of lines in the projective space with certain properties. Then, we employ the line-point incidence matrix and Johnson bounds for constant weig…
The locally repairable codes (LRCs) were introduced to correct erasures efficiently in distributed storage systems. LRCs are extensively studied recently. In this paper, we first deal with the open case remained in \cite{q} and derive an improved upper bound for the minimum distances of LRCs. We also give an explicit construction for LRCs attaining this bound. Secondly, we consider the constructions of LRCs with any locality and availability which have high code rate and minimum distance as large as possible. We give a graphical model for LRCs. By using the deep results from graph theory, we c…
Distributed and cloud storage systems are used to reliably store large-scale data. Erasure codes have been recently proposed and used in real-world distributed and cloud storage systems such as Google File System, Microsoft Azure Storage, and Facebook HDFS-RAID, to enhance the reliability. In order to decrease the repair bandwidth and disk I/O, a class of erasure codes called locally repairable codes (LRCs) have been proposed which have small locality compare to other erasure codes. Although LRCs have small locality, they have lower minimum distance compare to the Singleton bound. Hence, seeki…
We establish a duality result between linear index coding and Locally Repairable Codes (LRCs). Specifically, we show that a natural extension of LRCs we call Generalized Locally Repairable Codes (GLCRs) are exactly dual to linear index codes. In a GLRC, every node is decodable from a specific set of other nodes and these sets induce a recoverability directed graph. We show that the dual linear subspace of a GLRC is a solution to an index coding instance where the side information graph is this GLRC recoverability graph. We show that the GLRC rate is equivalent to the complementary index coding…
The locally repairable code (LRC) studied in this paper is an $[n,k]$ linear code of which the value at each coordinate can be recovered by a linear combination of at most $r$ other coordinates. The central problem in this work is to determine the largest possible minimum distance for LRCs. First, an integer programming based upper bound is derived for any LRC. Then by solving the programming problem under certain conditions, an explicit upper bound is obtained for LRCs with parameters $n_1>n_2$, where $n_1 = \left\lceil \frac{n}{r+1} \right\rceil$ and $n_2 = n_1 (r+1) - n$. Finally, an explic…
We consider the locally repairable codes (LRC), aiming at sequential recovering multiple erasures. We define the (n,k,r,t)-SLRC (Sequential Locally Repairable Codes) as an [n,k] linear code where any t'(>= t) erasures can be sequentially recovered, each one by r (2<=r<k) other code symbols. Sequential recovering means that the erased symbols are recovered one by one, and an already recovered symbol can be used to recover the remaining erased symbols. This important recovering method, in contrast with the vastly studied parallel recovering, is currently far from understanding, say, lacking code…
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…
In recent years, several classes of codes are introduced to provide some fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) play an important role. However, most known constructions are over large fields with sizes close to the code length, which lead to the systems computationally expensive. Due to this, binary LRCs are of interest in practice. In this paper, we focus on binary linear LRCs with disjoint repair groups. We first derive an explicit bound for the dimension k of such codes, which can be served as a…
When a node in a distributed storage system fails, it needs to be promptly repaired to maintain system integrity. While typical erasure codes can provide a significant storage advantage over replication, they suffer from poor repair efficiency. Locally repairable codes (LRCs) tackle this issue by reducing the number of nodes participating in the repair process (locality), at the cost of reduced minimum distance. In this paper, we study the tradeoff between locality and minimum distance of LRCs with local codes that have arbitrary distance requirements. Unlike existing methods where both the lo…
Locally repairable codes with locality $r$ ($r$-LRCs for short) were introduced by Gopalan et al. \cite{1} to recover a failed node of the code from at most other $r$ available nodes. And then $(r,δ)$ locally repairable codes ($(r,δ)$-LRCs for short) were produced by Prakash et al. \cite{2} for tolerating multiple failed nodes. An $r$-LRC can be viewed as an $(r,2)$-LRC. An $(r,δ)$-LRC is called optimal if it achieves the Singleton-type bound. It has been a great challenge to construct $q$-ary optimal $(r,δ)$-LRCs with length much larger than $q$. Surprisingly, Luo et al. \cite{3} presented a …
In this work it is shown that locally repairable codes (LRCs) can be list-decoded efficiently beyond the Johnson radius for a large range of parameters by utilizing the local error-correction capabilities. The corresponding decoding radius is derived and the asymptotic behavior is analyzed. A general list-decoding algorithm for LRCs that achieves this radius is proposed along with an explicit realization for LRCs that are subcodes of Reed--Solomon codes (such as, e.g., Tamo--Barg LRCs). Further, a probabilistic algorithm of low complexity for unique decoding of LRCs is given and its success pr…
Recently, locally repairable codes has gained significant interest for their potential applications in distributed storage systems. However, most constructions in existence are over fields with size that grows with the number of servers, which makes the systems computationally expensive and difficult to maintain. Here, we study linear locally repairable codes over the binary field, tolerating multiple local erasures. We derive bounds on the minimum distance on such codes, and give examples of LRCs achieving these bounds. Our main technical tools come from matroid theory, and as a byproduct of …
Locally repairable codes (LRCs) are a class of codes designed for the local correction of erasures. They have received considerable attention in recent years due to their applications in distributed storage. Most existing results on LRCs do not explicitly take into consideration the field size $q$, i.e., the size of the code alphabet. In particular, for the binary case, only a few results are known. In this work, we present an upper bound on the minimum distance $d$ of linear LRCs with availability, based on the work of Cadambe and Mazumdar. The bound takes into account the code length $n$, di…
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…
We construct an explicit family of locally repairable and locally regenerating codes whose existence was proven in a recent work by Kamath et al. about codes with local regeneration but no explicit construction was given. This explicit family of codes is based on HashTag codes. HashTag codes are recently defined vector codes with different vector length $α$ (also called a sub-packetization level) that achieve the optimal repair bandwidth of MSR codes or near-optimal repair bandwidth depending on the sub-packetization level. We applied the technique of parity-splitting code construction. We sho…
Recent years, several new types of codes were introduced to provide fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) have played an important role. A linear code is said to have locality $r$ if each of its code symbols can be repaired by accessing at most $r$ other code symbols. For an LRC with length $n$, dimension $k$ and locality $r$, its minimum distance $d$ was proved to satisfy the Singleton-like bound $d\leq n-k-\lceil k/r\rceil+2$. Since then, many works have been done for constructing LRCs meeting th…
As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the research of LRCs, bounds and constructions of the corresponding optimal codes are of particular concerns. In this work, codes with $(r,δ)$-locality which have optimal minimal distance w.r.t. the bound given by Prakash et al. \cite{Prakash2012Optimal} are considered. Through parity check matrix approach, constructions of both optimal $(r,δ)$-LRCs with all symbol l…
A linear block code with dimension $k$, length $n$, and minimum distance $d$ is called a locally repairable code (LRC) with locality $r$ if it can retrieve any coded symbol by at most $r$ other coded symbols. LRCs have been recently proposed and used in practice in distributed storage systems (DSSs) such as Windows Azure storage and Facebook HDFS-RAID. Theoretical bounds on the maximum locality of LRCs ($r$) have been established. The \textit{average} locality of an LRC ($\overline{r}$) directly affects the costly repair bandwidth, disk I/O, and number of nodes involved in the repair process o…
