[Paper Review] A Unified Form of EVENODD and RDP Codes and Their Efficient Decoding

Hanxu Hou, Yunghsiang S. Han+2 more|arXiv (Cornell University)|Mar 9, 2018

Advanced Data Storage Technologies23 references3 citations

TL;DR

This paper presents a unified construction of EVENODD and RDP codes as generalized MDS array codes with parity computed along polygonal lines, enabling a novel LU-based decoding algorithm over a simplified ring structure. The method reduces XOR operations by 20–78.5% compared to existing decoding methods, especially for more than three failed information columns, significantly improving decoding efficiency in high-reliability storage systems.

ABSTRACT

Array codes have been widely employed in storage systems, such as Redundant Arrays of Inexpensive Disks (RAID). The row-diagonal parity (RDP) codes and EVENODD codes are two popular double-parity array codes. As the capacity of hard disks increases, better fault tolerance by using array codes with three or more parity disks is needed. Although many extensions of RDP codes and EVENODD codes have been proposed, the high decoding complexity is the main drawback of them. In this paper, we present a new construction for all families of EVENODD codes and RDP codes, and propose a unified form of them. Under this unified form, RDP codes can be treated as shortened codes of EVENODD codes. Moreover, an efficient decoding algorithm based on an LU factorization of Vandermonde matrix is proposed when the number of continuous surviving parity columns is no less than the number of erased information columns. The new decoding algorithm is faster than the existing algorithms when more than three information columns fail. The proposed efficient decoding algorithm is also applicable to other Vandermonde array codes. Thus the proposed MDS array code is practically very meaningful for storage systems that need higher reliability.

Motivation & Objective

• To unify existing EVENODD and RDP codes into a single generalized framework for improved design flexibility and decoding efficiency.
• To address the high decoding complexity of extended EVENODD and RDP codes when more than three columns fail, especially in systems requiring triple or higher fault tolerance.
• To develop a low-complexity decoding algorithm applicable to all Vandermonde-based MDS array codes, particularly for multiple information column erasures.
• To demonstrate that the proposed decoding method outperforms existing algorithms like Blaum-Roth decoding in terms of XOR operation count.

Proposed method

• Proposes a unified code construction where RDP codes are treated as shortened versions of EVENODD codes under a generalized parity structure.
• Introduces a decoding method based on LU factorization of Vandermonde matrices over a simplified ring $ R_p $, avoiding costly operations over $ \mathbb{F}_2[x]/M_p(x) $.
• Reduces decoding complexity by solving Vandermonde systems over $ R_p $, then reducing results modulo $ M_p(x) $, which is more efficient than direct computation in the polynomial ring.
• Applies the LU decoding algorithm when the number of continuous surviving parity columns is at least the number of erased information columns, enabling efficient recovery.
• Uses a polygonal line pattern for parity computation, expanding design space for optimized recovery strategies.
• Employs systematic encoding and efficient erasure recovery via XOR-based operations, suitable for practical RAID and SSD storage systems.

Experimental results

Research questions

• RQ1 Can EVENODD and RDP codes be unified under a single generalized code structure that preserves their MDS property and decoding efficiency?
• RQ2 How can decoding complexity be reduced for multiple information column erasures in extended EVENODD and RDP codes?
• RQ3 Is it possible to achieve lower XOR operation counts in decoding by simplifying the algebraic structure used in Vandermonde matrix inversion?
• RQ4 Does the proposed LU decoding method outperform existing algorithms like Blaum-Roth decoding in terms of computational efficiency for multiple erasures?

Key findings

• The proposed LU decoding method reduces decoding complexity by 20.1% to 71.3% for $ r=4 $ and 20.4% to 68.5% for $ r=5 $ compared to the Blaum-Roth decoding method when $ p $ ranges from 5 to 59.
• For $ p=5 $ and $ r=5 $, the decoding complexity of RDP codes is reduced by 78.5% using the proposed method, demonstrating the largest improvement at small $ p $ and high $ r $.
• The decoding method is more efficient because it operates over $ R_p $ instead of $ \mathbb{F}_2[x]/M_p(x) $, reducing costly polynomial arithmetic operations.
• The unified code construction allows RDP codes to be viewed as shortened EVENODD codes, enabling a common framework for analysis and optimization.
• The method is applicable to all Vandermonde MDS array codes, making it broadly useful for future storage systems requiring high reliability.
• The proposed algorithm achieves lower decoding complexity than Cramer’s rule and existing interpolation-based methods, especially for more than three failed columns.