Search Results (6)

The rapid development of machine learning, large language models, and related technologies has greatly increased the demand for data storage capacity. Therefore, the role of distributed storage systems in such applications has become more prominent. However, it is inevitable that a single node fails in a distributed storage system during long-term use. Being able to repair failed nodes in a timely manner is extremely important for the stable operation of distributed storage systems, and a specific encoding scheme is required to meet the needs of efficiently repairing failed nodes. This research presents a novel family of binary locally repairable codes (LRCs) developed using multiple disjoint recovery sets constructed based on mutually orthogonal Latin squares (MOLS). The proposed constructions achieve distance optimality under the Singleton-like bound for LRCs with availability. Specifically, the codes are parameterized as $(n=r^2+tr,k=r^2,r,t)$ and $(n=rm+tm,k=rm,r,t)$, where n is the block length, k is the dimension, r is the locality, and t is the availability. These codes achieve minimum distance $d=t+1$, guaranteeing efficient recovery with t disjoint repair sets, each of size r. Compared to existing constructions, the proposed codes offer significant improvements in terms of code rate $\mathcal{R}={\displaystyle \frac{r}{r+t}}$, support for larger block lengths, and reduced finite field size requirements (field size $q=2$). Additionally, a method is introduced to improve the minimum distance of codes with even availability t, constructing codes with parameters $(n+1,k,r,t)$ and $d=t+2$, while preserving optimality. These properties make the proposed codes particularly suitable for distributed storage systems, where efficient and parallel repair of failed nodes is critical. Full article

Locally Repairable Codes (LRCs) have become the dominant design in wide-stripe erasure coding storage systems due to their excellent locality and low repair bandwidth. In such systems, the repair degree—defined as the number of helper nodes contacted during data recovery—is a key performance metric. However, as stripe width increases, the probability of multiple simultaneous node failures grows, which significantly raises the repair degree in traditional LRCs. Addressing this challenge, we propose a new family of codes called TFR-LRCs (Locally Repairable Codes for balancing fault tolerance and repair efficiency). TFR-LRCs introduce flexible design choices that allow trade-offs between fault tolerance and repair degree: they can reduce the repair degree by slightly increasing storage overhead, or enhance fault tolerance by tolerating a slightly higher repair degree. We design a matrix-based construction to generate TFR-LRCs and evaluate their performance through extensive simulations. The results show that, under multiple failure scenarios, TFR-LRC reduces the repair degree by up to 35% compared with conventional LRCs, while preserving the original LRC structure. Moreover, under identical code parameters, TFR-LRC achieves improved fault tolerance, tolerating up to $g+2$ failures versus $g+1$ in conventional LRCs, with minimal additional cost. Notably, in maintenance mode, where entire racks may become temporarily unavailable, TFR-LRC demonstrates substantially better recovery efficiency compared to existing LRC schemes, making it a practical choice for real-world deployments. Full article

In this paper, we generalize the construction of locally repairable codes (LRCs) by leveraging pairwise balanced designs (PBDs) and balanced incomplete block designs (BIBDs) to construct codes with nonuniform locality or nonuniform availability. Our constructions prioritize binary implementations for practical deployment while achieving optimal or near-optimal performance in terms of rate, minimum distance, and repair efficiency. Specifically, we propose distance-optimal LRCs with nonuniform localities and message-symbol $(r,t)$-availability. These binary constructions achieve optimal minimum distance under known bounds and have higher code rates than existing works. We also address open problems in the literature, including constructions where $r\nmid k$, and demonstrate that our constructions encompass or outperform several prior works. Full article

This paper summarizes the details of recently proposed binary locally repairable codes (BLRCs) and their features. The construction of codes over a small alphabet size of symbols is of particular interest for efficient hardware implementation. Therefore, BLRCs are highly noteworthy because no multiplication is required during the encoding, decoding, and repair processes. We explain the various construction approaches of BLRCs such as cyclic code based, bipartite graph based, anticode based, partial spread based, and generalized Hamming code based techniques. We also describe code generation methods based on modifications for linear codes such as extending, shorting, expurgating, and augmenting. Finally, we summarize and compare the parameters of the discussed constructions. Full article

Locally repairable codes (LRCs) are a new family of erasure codes used in distributed storage systems which have attracted a great deal of interest in recent years. For an $[n,k,d]$ linear code, if a code symbol can be repaired by t disjoint groups of other code symbols, where each group contains at most r code symbols, it is said to have availability- $(r,t)$ . Single-parity LRCs are LRCs with a constraint that each repairable group contains exactly one parity symbol. For an $[n,k,d]$ single-parity LRC with availability- $(r,t)$ for the information symbols (single-parity LRCs), the minimum distance satisfies $d\le n-k-\lceil kt/r\rceil +t+1$ . In this paper, we focus on the study of single-parity LRCs with availability- $(r,t)$ for information symbols. Based on the standard form of generator matrices, we present a novel characterization of single-parity LRCs with availability $t\ge 1$ . Then, a simple and straightforward proof for the Singleton-type bound is given based on the new characterization. Some necessary conditions for optimal single-parity LRCs with availability $t\ge 1$ are obtained, which might provide some guidelines for optimal coding constructions. Full article

In this paper, a new family of binary LRCs (BLRCs) with locality 2 and uneven availabilities for hot data is proposed, which has a high information symbol availability and low parity symbol availabilities for the local repair of distributed storage systems. The local repair of each information symbol for the proposed codes can be done not by accessing other information symbols but only by accessing parity symbols. The proposed BLRCs with $k=4$ achieve the optimality on the information length for their given code length, minimum Hamming distance, locality, and availability in terms of the well-known theoretical upper bound. Full article