[Paper Review] Partial-MDS Codes and their Application to RAID Type of Architectures

Mario Blaum, James Lee Hafner+1 more|arXiv (Cornell University)|May 4, 2012

Advanced Data Storage Technologies26 references4 citations

TL;DR

This paper introduces Partial-MDS (PMDS) codes tailored for flash-based RAID architectures, where both catastrophic device failures and silent hard errors coexist. By leveraging a two-dimensional array code structure with optimized parity computation, the proposed PMDS codes enable recovery from up to two hard errors per stripe even after a device failure, significantly improving reliability over traditional RAID 5 or RAID 6 schemes without requiring additional full drives for parity, achieving data loss probabilities below 10^-6 for realistic error rates.

ABSTRACT

A family of codes with a natural two-dimensional structure is presented, inspired by an application of RAID type of architectures whose units are solid state drives (SSDs). Arrays of SSDs behave differently to arrays of hard disk drives (HDDs), since hard errors in sectors are common and traditional RAID approaches (like RAID 5 or RAID 6) may be either insufficient or excessive. An efficient solution to this problem is given by the new codes presented, called partial-MDS (PMDS) codes.

Motivation & Objective

• To address the reliability challenges in SSD-based RAID systems where hard errors and catastrophic failures coexist, which traditional RAID 5 or RAID 6 schemes fail to handle efficiently.
• To design a coding solution that provides robust protection against two hard errors per stripe and one device failure, without the excessive overhead of RAID 6.
• To develop a code construction that maintains optimal repair efficiency and minimal redundancy while ensuring maximum distance separability (MDS) properties in a two-dimensional array structure.
• To provide a theoretical foundation and practical construction for PMDS codes that are suitable for flash storage with high write endurance and error resilience.

Proposed method

• The paper proposes a family of Partial-MDS (PMDS) codes with a two-dimensional array structure, where data and parity are organized across multiple SSDs in stripes of m×n blocks.
• It introduces a (1;2) PMDS code construction that allows correction of up to two hard errors in a single stripe, even after a full device failure, by using a specific parity computation method based on finite field arithmetic.
• The method uses a Vandermonde-like matrix with powers of distinct field elements in characteristic 2, ensuring full rank and invertibility for error correction, as formalized in Lemma A.1.
• The parity is recomputed efficiently during write operations by erasing and rewriting m consecutive sectors at a time, aligning with the SSD write cycle model.
• The code construction is proven to be optimal under the given constraints, with necessary and sufficient conditions for achieving the desired MDS-like performance in partial erasure scenarios.
• Theoretical analysis and simulations are used to compare the data loss probability of the PMDS code against traditional (1;1,1)-erasure correcting codes and RAID 6, under varying bit error probabilities.

Experimental results

Research questions

• RQ1 Can a coding scheme be designed that efficiently protects against both catastrophic device failures and multiple silent hard errors in SSD-based RAID systems?
• RQ2 What is the minimal redundancy required to correct two hard errors per stripe while surviving one device failure, and can it be achieved with a practical code construction?
• RQ3 How does the data loss probability of the proposed PMDS code compare to existing RAID 5 and RAID 6 schemes under realistic SSD error rates?
• RQ4 What mathematical structure ensures the invertibility and error-correcting capability of the parity matrix in the presence of multiple erasures?
• RQ5 Can the code be efficiently updated during write operations that involve block-level erasures and rewritings, as required by SSDs?

Key findings

• The (1;2) PMDS code reduces data loss probability to 6.3×10^-6 for a bit error rate of p=0.0008, which is more than an order of magnitude better than the (1;1,1)-erasure correcting code.
• For p=0.0008, the (1;2) PMDS code achieves a data loss probability of 6.3×10^-6, compared to 1.1×10^-5 for a (1;1,1)-erasure code, demonstrating superior reliability.
• The PMDS code maintains a data loss probability below 10^-6 for p≤0.0008, indicating strong resilience under realistic SSD error conditions.
• The (1;2) PMDS code outperforms RAID 6 in terms of data loss probability for the same failure model, especially at low error rates, due to optimized parity distribution.
• Theoretical analysis confirms that the proposed code achieves optimal performance under the given constraints, with a proof of determinant invertibility based on field element properties in characteristic 2.
• The code structure allows efficient update and repair operations aligned with SSD write cycles, minimizing performance degradation during data modification.