For an [n, k, d] (r, δ)-locally repairable codes ((r, δ)-LRCs), its minimum distance d satisfies the Singleton-like bound. The construction of optimal (r, δ)-LRC, attaining this Singleton-like bound, is an important research problem in recent years for thier applications in distributed storage systems. In this letter, we use Reed-Solomon codes to construct two classes of optimal (r, δ)-LRCs. The optimal LRCs are given by the evaluations of multiple polynomials of degree at most r - 1 at some points in Fq. The first class gives the [(r + δ - 1)t, rt - s, δ + s] optimal (r, δ)-LRC over Fq provided that r + δ + s - 1≤q, s≤δ, s

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.

Locally repairable codes have attracted lots of interest in Distributed Storage Systems. If a symbol of a code can be repaired respectively by t disjoint groups of other symbols, each groups has size at most r, we say that the code symbol has (r, t)-locality. In this paper, we employ parity-check matrix to construct information single-parity (r, t)-locality LRCs. All our codes attain the Singleton-like bound of LRCs where each repair group contains a single parity symbol and thus are optimal.

Locally repairable codes (LRCs) with availability have received considerable attention in recent years since they are able to solve many problems in distributed storage systems such as repairing multiple node failures and managing hot data. Constructing LRCs with locality r and availability t (also called (r, t)-LRCs) with new parameters becomes an interesting research subject in coding theory. The objective of this paper is to propose two generic constructions of cyclic (r, t)-LRCs via linearized polynomials over finite fields. These two constructions include two earlier ones of cyclic LRCs from trace functions and truncated trace functions as special cases and lead to LRCs with new parameters that can not be produced by earlier ones.

Locally repairable codes (LRCs) are a type of new erasure codes designed for modern distributed storage systems (DSSs). In order to obtain ternary LRCs of distance 6, firstly, we propose constructions with disjoint repair groups and construct several families of LRCs with 1 ≤ r ≤ 6, where codes with 3 ≤ r ≤ 6 are obtained through a search algorithm. Then, we propose a new method to extend the length of codes without changing the distance. By employing the methods such as expansion and deletion, we obtain more LRCs from a known LRC. The resulting LRCs are optimal or near optimal in terms of the Cadambe-Mazumdar (C-M) bound.

Locally repairable codes (LRCs) are implemented in distributed storage systems (DSSs) due to their low repair overhead. The locality of an LRC is the number of nodes in DSSs that participate in the repair of failed nodes, which characterizes the repair cost. An LRC is called optimal if its minimum distance attains the Singleton-type upper bound [1]. In this letter, optimal LRCs are considered. Using the concept of projective code in projective space PG(k, q) and shortening strategy, LRCs with d=3 are proposed. Meantime, derived from an ovoid [q2+1, 4, q2]q code (responding to a maximal (q2+1)-cap in PG(3, q)), optimal LRCs over Fq with d=4 are constructed.

The Gabidulin-based locally repairable code (LRC) construction by Silberstein et al. is an important example of distance optimal (r,δ)-LRCs. Its distance optimality has been further shown to cover the case of multiple (r,δ)-locality, where the (r,δ)-locality constraints are different among different symbols. However, the optimality only holds under the ordered (r,δ) condition, where the parameters of the multiple (r,δ)-locality satisfy a specific ordering condition. In this letter, we show that Gabidulin-based LRCs are still distance optimal even without the ordered (r,δ) condition.

In this paper, we investigate how to obtain binary locally repairable codes (LRCs) with good locality and availability from binary Simplex codes. We first propose a Combination code having the generator matrix with all the columns of positive weights less than or equal to a given value. Such a code can be also obtained by puncturing all the columns of weights larger than a given value from a binary Simplex Code. We call by block-puncturing such puncturing method. Furthermore, we suggest a heuristic puncturing method, called subblock-puncturing, that punctures a few more columns of the largest weight from the Combination code. We determine the minimum distance, locality, availability, joint information locality, joint information availability of Combination codes in closed-form. We also demonstrate the optimality of the proposed codes with certain choices of parameters in terms of some well-known bounds.

Locally repairable codes, which can repair erased symbols from other symbols, have attracted a good deal of attention in recent years because its local repair property is effective on distributed storage systems. (ru, δu)u∈[s]-locally repairable codes with multiple localities, which are an extension of ordinary locally repairable codes, can repair δu-1 erased symbols simultaneously from a set consisting of at most ru symbols. An upper bound on the minimum distance of these codes and a construction method of optimal codes, attaining this bound with equality, were given by Chen, Hao, and Xia. In this paper, we discuss the parameter restrictions of the existing construction, and we propose explicit constructions of optimal codes with multiple localities with relaxed restrictions based on the encoding polynomial introduced by Tamo and Barg. The proposed construction can design a code whose minimum distance is unrealizable by the existing construction.

Locally repairable code (LRC) can recover any codeword symbol failure by accessing a small number of other symbols, which can increase the efficiency during the repair process. In a distributed storage system with locally repairable codes, any node failure can be rebuilt by accessing other fixed nodes. It is a promising prospect for the application of LRC. In this paper, some methods of constructing matrices which can generate codes with small locality will be proposed firstly. By analyzing the parameters, we construct the generator matrices of the best-known ternary linear codes of dimension 6, using methods such as shortening, puncturing and expansion. After analyzing the linear dependence of the column vectors in the generator matrices above, we find out the locality of the codes they generate. Many codes with small locality have been found.

Locally repairable codes have recently been applied in distributed storage systems because of their excellent local erasure-correction capability. A locally repairable code is a code with locality r, where each code symbol can be recovered by accessing at most r other code symbols. In this paper, we study the existence and construction of binary cyclic codes with locality 2. An overview of best binary cyclic LRCs with length 7≤n≤87 and locality 2 are summarized here.

Locally repairable codes (LRCs) have attracted much interest recently due to their applications in distributed storage systems. 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. An (n,k,r) LRC with locality r for the information symbols has minimum distance d≤n-k-⌈k/r⌉+2. In this letter, we study single-parity LRCs where every repair group contains exactly one parity symbol. Firstly, we give a new characterization of single-parity LRCs based on the standard form of generator matrices. For the optimal single-parity LRCs meeting the Singleton-like bound, we give necessary conditions on the structures of generator matrices. Then we construct all the optimal binary single-parity LRCs meeting the Singleton-like bound d≤n-k-⌈k/r⌉+2.

In this paper, we examine the locality property of the original Fractional Repetition (FR) codes and propose two constructions for FR codes with better locality. For this, we first derive the capacity of the FR codes with locality 2, that is the maximum size of the file that can be stored. Construction 1 generates an FR code with repetition degree 2 and locality 2. This code is optimal in the sense of achieving the capacity we derived. Construction 2 generates an FR code with repetition degree 3 and locality 2 based on 4-regular graphs with girth g. This code is also optimal in the same sense.