Optimal Locally Repairable Codes Research Articles

Singleton-optimal LRCs and perfect LRCs via cyclic and constacyclic codes

Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair costs by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic and constacyclic codes provide significant benefits of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic and constacyclic codes with long code lengths. Specifically, we first obtain two classes of q-ary cyclic Singleton-optimal (n,k,d=6;r=2)-LRCs with length n=3(q+1) when 3|(q−1) and q is even, and length n=32(q+1) when 3|(q−1) and q≡1(mod4), respectively. To the best of our knowledge, this is the first construction of q-ary cyclic Singleton-optimal LRCs with length n>q+1 and minimum distance d≥5. On the other hand, an LRC achieving the Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic codes, we construct two new families of q-ary perfect LRCs with length n=qm−1q−1, minimum distance d=5 and locality r=2.

Construction of Singleton-Type Optimal LRCs from Existing LRCs and Near-MDS Codes

Modern large scale distributed storage systems play a central role in data center and cloud storage, while node failure in data center is common. The lost data in failure node must be recovered efficiently. Locally repairable codes (LRCs) are designed to solve this problem. The locality of an LRC is the number of nodes that participate in recovering the lost data from node failure, which characterizes the repair efficiency. An LRC is called optimal if its minimum distance attains Singleton-type upper bound [1]. In this paper, using basic techniques of linear algebra over finite field, infinite optimal LRCs over extension fields are derived from a given optimal LRC over base field (or small field). Next, this paper investigates the relation between near-MDS codes with some constraints and LRCs, further, proposes an algorithm to determine locality of dual of a given linear code. Finally, based on near-MDS codes and the proposed algorithm, those obtained optimal LRCs are shown.

A characterization of optimal locally repairable codes

It is significant to investigate locally repairable codes (LRCs) since they have abundant applications to distributed storage systems. A locally repairable code is called optimal if it achieves the Singleton-like bound. In [27], I. Tamo et al. presented sufficient and necessary conditions for optimal LRCs. However, it is difficult to check these conditions for many parameters. In this work, we show that construction of optimal LRCs is equivalent to construction of a class of matrices with some conditions. With the help of these conditions, we completely determine all the optimal LRCs for some small parameters. In addition, by shortening technique for the matrices with these conditions, we can easily obtain many new optimal LRCs remaining the minimum distance.

A classification for optimal quaternary locally repairable codes

In 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) 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 studies have been done for constructing LRCs meeting the Singleton-like bound over small fields. In this paper, we study quaternary LRCs meeting Singleton-like bound through a parity-check matrix approach, using tools from combinatorial designs and finite geometry. From this, we prove that there are $27$ different classes of parameters for optimal quaternary LRCs. Moreover, for each class, explicit constructions of corresponding optimal quaternary LRCs are presented. In addition, using tools from finite geometry, this paper also introduces a new method to determine the existence of the optimal LRCs.

On binary locally repairable codes with distance four

We propose a unified construction for binary locally repairable codes (LRCs) with distance four. When there is a binary linear code [n,k,4] without all zero coordinates, we can construct a binary LRC [n,k,4] with very good locality r. Conditions for which these LRCs attain the Cadambe-Mazumdar bound are also presented. We prove that all our LRCs with locality r≤3 are optimal LRCs, and most of our LRCs with locality r≥4 can achieve the Cadambe-Mazumdar bound. Especially, if n≤256, all the constructed distance optimal [n,k,4] LRCs, except 17 codes, can achieve the Cadambe-Mazumdar bound. We also show that six known constructions are covered in our construction, and eight known constructions can be improved by our construction.

Constructions of Optimal Codes With Hierarchical Locality

Locally repairable codes (LRCs) and codes with hierarchical locality (H-LRCs) have a great significance due to their applications in distributed storage systems. Constructing LRCs or H-LRCs achieving the Singleton-type bound is a challenging task and has received much attention. In this paper, we make use of optimal LRCs to construct a family of optimal H-LRCs. For this purpose we first construct a new class of optimal LRCs via generalized Reed-Solomon codes (GRS codes). Next, based on these optimal LRCs we present some new optimal H-LRCs, of which the parameters are more flexible than those given in [1] and [18].

Constructions of ternary LRCs with optimal distance

As a new class of erasure codes, locally repairable codes (LRCs) can provide data repair capability for distributed storage system. In this paper, we firstly propose an effective construction method to obtain optimal LRCs based on ternary optimal codes. Then, three classes of optimal LRCs are constructed by ternary cyclic codes and constant cyclic codes. All codes we gained have the optimal distance and reach the Cadambe-Mazumdar (C-M) bound.

Two classes of optimal LRCs with information (r, t)-locality

Locally repairable codes (LRCs) with (r, t)-locality have received considerable attention in recent years, since they are able to solve common problems in distributed storage systems such as repairing multiple node failures and managing hot data. Constructing LRCs with excellent parameters becomes an interesting research subject in distributed storage systems and coding theory. In this paper, we present two constructions of LRCs with information (r, t)-locality based on linear algebra and partial geometry, respectively. Both constructions generate LRCs with new parameters which are optimal with respect to the bound proposed by Rawat et al. (IEEE Trans Inf Theory 62(8):4481–4493, 2016).

Optimal (r, δ)-Locally Repairable Codes From Simplex Code and Cap Code

Locally repairable codes (LRCs) are implemented in distributed storage systems (DSSs) due to their low repair overhead. A linear code $\mathcal {C}$ is said to have $(r,\delta)$ -locality if for each coordinate $i$ , there exists a punctured subcode of $\mathcal {C}$ with support containing $i$ , whose length is at most $r+\delta -1$ , and whose minimum distance is at least $\delta $ . An LRC is called optimal if its minimum distance attains Singleton-type bound was proposed. In this letter, optimal LRCs are considered. We first determine $(r,\delta)$ -locality of three dimensional Simplex code, then using anticode strategy, a class of $[{3q,3,2q-1}]_{q}$ LRCs with $(2,q)$ locality are derived for general $q$ . Finally, using an ovoid in $PG(3, q)$ , we construct $[q^{2}+1,4,q(q-1)]_{q}$ and $[{4q-4,4,3q-5}]_{q}$ LRCs with $r=3$ and $\delta =q-1$ . All LRCs constructed in this letter attain the Singleton-type bound.

Optimal Locally Repairable Codes for Parallel Reading

Locally repairable codes (LRCs) have important applications in distributed storage systems. In this paper, we study q-ary [n, k, d] LRCs with (r, t, δ)-information-locality, where each of the i-th (1 ≤ i ≤ k) information symbol is contained in t punctured subcodes with length ≤ r + δ - 1, minimum distance δ, and the i-th information symbol is the unique common code symbol of these t subcodes, furthermore, each subcode contains exactly δ - 1 parity symbols. Firstly, an upper bound on the minimum distance of such q-ary LRCs with (r, t, δ)-information-locality is given. Then, we propose a general construction framework of q-ary optimal LRCs with (r, t, δ)-information-locality and minimum distanced = t(δ-1)+1, where the required field size is just q ≥ r+δ-2. The proposed optimal LRCs can always repair a failed information node locally in case of at most tδ - 1 node failures. Moreover, multiple repair subcodes can support parallel readings of data, thus make the proposed codes attractive for distributed storage systems with hot data.

Constructions of Optimal $(r,\delta)$ Locally Repairable Codes via Constacyclic Codes

Locally repairable codes (LRCs) are introduced in distributed storage systems due to their low repair overhead. An LRC is called optimal if its minimum distance attains the Singleton-like upper bound. Chen et al. (2018) recently studied the constructions of optimal $(r, \delta)$ -LRCs with length $n \mid (q+1)$ and $(r+\delta -1) \mid n$ , where many classes of optimal cyclic constructions were obtained. In this paper, by employing constacyclic MDS codes, we construct seven classes of optimal $(r, \delta)$ -LRCs with new parameters. After adding these new optimal LRCs via constacyclic codes, we have completely obtained all optimal $(r, \delta)$ -LRCs with length $n \mid (q+1)$ and $(r+\delta -1) \mid n$ for all possible parameters for the completeness in the coding theory. It is worth noting that the optimal constacyclic LRCs with new parameters provide more alternatives to cyclic LRCs in the practical demands of distributed storage systems, where specific values of $n$ , $k$ , $r$ , and $\delta $ are required. Moreover, constacyclic LRCs also possess the encoding and decoding efficiency as cyclic LRCs.

Complete Characterizations of Optimal Locally Repairable Codes With Locality 1 and $K-1$

A locally repairable code (LRC) is a [n, k, d] linear code with length n, dimension k, minimum distance d and locality r, which means that every code symbol can be repaired by at most r other symbols. LRCs have become an important candidate in distributed storage systems due to their relatively low I/O cost. An LRC is said to be optimal if its minimum distance meets one of the Singleton-like bounds. This paper considers the optimal constructions of LRCs with locality r = 1 and r = k - 1, which involves three types: r-local LRCs, (r, δ)-LRCs and LRCs with t-availability. Specifically, we first prove that the existence of an optimal LRC with locality r = 1 is equivalent to that of an MDS code with certain parameters. Thus we can completely characterize the three types of optimal LRCs with r = 1 based on some known constructions of MDS codes. Near MDS codes is a special class of sub-optimal linear code whose minimum distance d = n - k and the i-th generalized Hamming weight achieves the generalized Singleton bound for 2 ≤ i ≤ k. For r = k - 1, we have established the connections between optimal r-local LRCs/ LRCs with t-availability and near MDS codes. Such connections can help to construct optimal LRCs with r = k - 1 from some known classes of near MDS codes.

Locally Repairable Codes

Distributed storage systems for large-scale applications typically use replication for reliability. Recently, erasure codes were used to reduce the large storage overhead, while increasing data reliability. A main limitation of off-the-shelf erasure codes is their high-repair cost during single node failure events. A major open problem in this area has been the design of codes that: 1) are repair efficient and 2) achieve arbitrarily high data rates. In this paper, we explore the repair metric of locality, which corresponds to the number of disk accesses required during a single node repair. Under this metric, we characterize an information theoretic tradeoff that binds together the locality, code distance, and storage capacity of each node. We show the existence of optimal locally repairable codes (LRCs) that achieve this tradeoff. The achievability proof uses a locality aware flow-graph gadget, which leads to a randomized code construction. Finally, we present an optimal and explicit LRC that achieves arbitrarily high data rates. Our locality optimal construction is based on simple combinations of Reed-Solomon blocks.