Optimal Locally Repairable Linear Codes

Abstract

Abstract-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.

Construction of New Fractional Repetition Codes from Relative Difference Sets with λ=1

“…We cannot find any existing FR code whose parameters are the same as the proposed ones, and thus, we plot the MBR capacity and the FR capacity bound φ(k) in (3). We can see that the (8,3,3) and (9,3,3) FR codes are optimal because they achieve the FR capacity bound for all k. The (15,4,4), (16,4,4), (24,5,5) and (25,5,5) FR codes do not exactly achieve the FR capacity bound, but the gaps become smaller; thus, it deserves to be called FR capacity-approaching. (15,4,4) and (16,4,4) FR codes; (c) (24,5,5) and (25,5,5) FR codes.…”

“…In the following examples, we provide incidence matrices for FR codes with parameters (9, 3, 3), (16,4,4) and (25,5,5). …”

“…The incidence matrices of the proposed FR codes satisfy the above condition for a given k, and thus, M(k) of the proposed FR codes is larger than or equal to the MBR capacity. Figure 2 demonstrates the number of data symbols M(k) of the proposed FR codes with parameters (8,3,3), (9,3,3), (15,4,4), (16,4,4), (24,5,5) and (25,5,5). We cannot find any existing FR code whose parameters are the same as the proposed ones, and thus, we plot the MBR capacity and the FR capacity bound φ(k) in (3).…”

“…Traditional solutions such as simple triplication and Reed-Solomon (RS) codes are no longer enough to efficiently maintain DSSs and enhance the reliability of stored data because, for node failure-handling, we have to consider the tremendous amount of data traffic over the network in DSS, the number of disk I/O (input/output) and the availability of local data processing, as well as the redundancy of stored data. Thus, it is necessary to find a new class of node failure-handling protocols that is well-fitted for the DSS environment, and for this reason, locally repairable codes [3][4][5][6][7] and regenerating codes [8] have recently attracted much attention.…”

Construction of Fractional Repetition Codes with Variable Parameters for Distributed Storage Systems

“…Rather than using a random protocol, we need to adopt some mathematical structures in storing big data to minimize the overall cost of DSSs. Recently, locally repairable codes (LRCs) [3][4][5][6][7], and regenerating codes (RCs) [8][9][10][11] have been proposed for DSSs to support cost-reducing repair against node failures as well as data reconstruction. For repair, LRCs aim to connect to a small number of other nodes while RCs aim to use a small amount of network bandwidth.…”