Explicit constructions of high-rate MDS array codes with optimal repair bandwidth (1604.00454v2)

Abstract: Maximum distance separable (MDS) codes are optimal error-correcting codes in the sense that they provide the maximum failure-tolerance for a given number of parity nodes. Suppose that an MDS code with $k$ information nodes and $r=n-k$ parity nodes is used to encode data in a distributed storage system. It is known that if $h$ out of the $n$ nodes are inaccessible and $d$ surviving (helper) nodes are used to recover the lost data, then we need to download at least $h/(d+h-k)$ fraction of the data stored in each of the helper nodes (Dimakis et. al., 2010 and Cadambe et al., 2013). If this lower bound is achieved for the repair of any $h$ erased nodes from any $d$ helper nodes, we say that the MDS code has the $(h,d)$-optimal repair property. We study high-rate MDS array codes with the optimal repair property. Explicit constructions of such codes in the literature are only available for the cases where there are at most 3 parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In this paper, given any $r$ and $n$, we present two explicit constructions of MDS array codes with the $(h,d)$-optimal repair property for all $h\le r$ and $k\le d\le n-h$ simultaneously. Codes in the first family can be constructed over any base field $F$ as long as $|F|\ge sn,$ where $s=\text{lcm}(1,2,\dots,r).$ The encoding, decoding, repair of failed nodes, and update procedures of these codes all have low complexity. Codes in the second family have the optimal access property and can be constructed over any base field $F$ as long as $|F|\ge n+1.$ Moreover, both code families have the optimal error resilience capability when repairing failed nodes. We also construct several other related families of MDS codes with the optimal repair property.

• The paper introduces explicit MDS array code constructions that attain optimal repair bandwidth for any number of parity nodes and code lengths.
• It leverages linear combinations over finite fields to enable universally error-resilient repairs with reduced computational complexity.
• The research reduces the required base field size and supports versatile repair scenarios, enhancing efficiency in distributed storage systems.

Overview of High-Rate MDS Array Codes with Optimal Repair Bandwidth

Introduction and Background

The paper presented focuses on the construction of high-rate Maximum Distance Separable (MDS) array codes optimized for repair bandwidth, a critical aspect in distributed storage systems. MDS codes are paramount due to their optimal failure tolerance relative to the number of parity nodes. The specific problem addressed is the efficient repair of data following node failures, minimizing the repair bandwidth required for these corrections. The authors provide explicit constructions of such codes through sophisticated combinatorial techniques and linear algebra within finite fields.

Key Contributions

The main contributions of this research include:

1. Explicit Construction: The paper provides two explicit families of MDS array codes that achieve optimal repair bandwidth for any number of parity nodes $r$ and code length $n$. This surpasses prior limitations where such constructions were only feasible with up to 3 parity nodes and restricted repair scenarios.
2. Innovative Use of Finite Fields: The authors leverage linear combinations of finite field elements to construct codes that are capable of universally error-resilient (UER) repairs. This approach ensures that during repairs, the number of downloaded symbols meet the lower bounds established in seminal works by Dimakis et al., and Cadambe et al.
3. Reduction of Field Size Requirements: The construction reduces the required size of the base field compared to previous works, demanding only $|F| \geq sn$, where $s$ is determined by the least common multiple of integers up to $r$.
4. Support for Multiple Repair Scenarios and Reduced Complexity: The codes devised facilitate the repair of any $h$ failed nodes using any subset of $d$ helper nodes, simultaneously minimizing bandwidth across various $(h,d)$-combinations. This versatility is combined with low-complexity encoding, decoding, and update procedures using operations on small matrix sizes.
5. Optimal Access Property: One of the proposed constructions uniquely ensures optimal access, meaning the repair from node failures is accomplished by directly accessing the minimum amount of data.

Numerical and Theoretical Implications

The explicit constructions demonstrate strong numerical properties, achieving optimal repair bandwidth by allowing downloads from helper nodes at either $h/(d+h-k)$ or $d/(d+h-k)$ fractions. The theoretical implications extend to the structure and implementation of large-scale storage systems where bandwidth and speed are crucial. The reduction in required field size and computational complexity further enhance this practical applicability.

Speculation on Future Developments

The advancements in this paper set a foundation for further research into optimizing distributed storage systems, particularly in exploring the potential of creating similar explicit constructions over substantially smaller fields. Moreover, the techniques might inspire new methods in data compression and transmission, given their ability to minimize redundancies and maximize efficiency.

Conclusion

This paper offers significant advancements in the field of coding theory, providing comprehensive solutions to high-rate MDS array codes with optimal repair bandwidth. By presenting explicit constructions with reduced complexity and field size requirements, the authors have refined the mathematical framework within which efficient data storage and retrieval can be executed. These contributions hold promise for both theoretical advancements and practical applications in data management and distributed computing environments.