X-Code: MDS Array Codes with Optimal Encoding

2013

ABSTRACT Maximum-distance separable (MDS) array codes with high rate and an optimal repair property were introduced recently. These codes could be applied in distributed storage systems, where they minimize the communication and disk access required for the recovery of failed nodes. However, the encoding and decoding algorithms of the proposed codes use arithmetic over finite fields of order greater than 2, which could result in a complex implementation. In this work, we present a construction of 2-parity MDS array codes, that allow for optimal repair of a failed information node using XOR operations only. The reduction of the field order is achieved by allowing more parity bits to be updated when a single information bit is being changed by the user.

2011 Information Theory and Applications Workshop, 2011

In this article we describe a class of error control codes called "diff-MDS" codes that are custom designed for highly resilient computer memory storage. The error scenarios of concern range from simple single bit errors, to memory chip failures and catastrophic memory module failures. Our approach to building codes for this setting relies on the concept of expurgating a parity code that is easy to decode for memory module failures so that a few additional small errors can be handled as well, thus preserving most of the decoding complexity advantages of the original code while extending its original intent. The manner in which we expurgate is carefully crafted so that the strength of the resulting code is comparable to that of a Reed-Solomon code when used for this particular setting. An instance of this class of algorithms has been incorporated in IBM's zEnterprise mainframe offering, setting a new industry standard for memory resiliency.

IEEE Transactions on Computers, 2005

This paper presents a class of binary maximum distance separable (MDS) array codes for tolerating disk failures in redundant arrays of inexpensive disks (RAID) architecture based on circular permutation matrices. The size of the information part is m×n, the size of the parity-check part is m×3, and the minimum distance is 4, where n is the number of information disks, the number of parity-check disks is 3, and (m+1) is a prime integer. In practical applications, m can be very large and n is from 20 to 50. The code rate is R=n/(n+3). These codes can be used for tolerating three disk failures. The encoding and decoding of the Reed-Solomon-like codes are very fast. There need to be 3mn XOR operations for encoding and (3mn+9(m+1)) XOR operations for decoding.

IEEE Transactions on Information Theory, 2013

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.

IEEE Transactions on Communications, 2006

In general, an (n, k) MDS code over GF(p m ) cannot correct all burst erasures of length (nk)m over GF(p). In this paper, we constructively show that such a linear MDS code over GF(p m ) can be modified to correct all burst erasures of length up to (n-k)m over GF(p), while maintaining its MDS property over GF(p m ).

2009

Three new families of lowest density maximum-distance separable (MDS) array codes are constructed, which are cyclic or quasi-cyclic. In addition to their optimal redundancy (MDS) and optimal update complexity (lowest density), the symmetry offered by the new codes can be utilized for simplified implementation in storage applications. The proof of the code properties has an indirect structure: first MDS codes that are not cyclic are constructed, and then transformed to cyclic codes by a minimum-distance preserving transformation.

arXiv (Cornell University), 2023

A code of length n is said to be (combinatorially) (ρ, L)-list decodable if the Hamming ball of radius ρn around any vector in the ambient space does not contain more than L codewords. We study a recently introduced class of higher order MDS codes, which are closely related (via duality) to codes that achieve a generalized Singleton bound for list decodability. For some ℓ ≥ 1, higher order MDS codes of length n, dimension k, and order ℓ are denoted as (n, k)-MDS(ℓ) codes. We present a number of results on the structure of these codes, identifying the 'extend-ability' of their parameters in various scenarios. Specifically, for some parameter regimes, we identify conditions under which (n 1 , k 1)-MDS(ℓ 1) codes can be obtained from (n 2 , k 2)-MDS(ℓ 2) codes, via various techniques. We believe that these results will aid in efficient constructions of higher order MDS codes. We also obtain a new field size upper bound for the existence of such codes, which arguably improves over the best known existing bound, in some parameter regimes.

ArXiv, 2008

In this paper, we describe a procedure for constructing $q$--ary $[N,3,N-2]$--MDS codes, of length $N\leq q+1$ (for $q$ odd) or $N\leq q+2$ (for $q$ even), using a set of non--degenerate Hermitian forms in $PG(2,q^2)$.

IEEE Access, 2021

Symbol-pair codes are used to protect against symbol-pair errors in high density data storage systems. One of the most important tasks in symbol-pair coding theory is to design MDS codes. MDS symbol-pair codes are optimal in the sense that such codes attain the Singleton bound. In this paper, a new class of MDS symbol-pair codes with code-length 5p and optimal pair distance of 7 is established. It is shown that for any prime p ≡ 1 (mod 5), we can always construct four p-ary MDS symbol-pair cyclic codes of length 5p of largest possible pair distance 7. We also completely determined all MDS symbol-pair and MDS b-symbol codes of length p s and 2p s over F p m + uF p m by filling in some missing cases, and rectifying some gaps in Type 3 codes of recent papers. As an applications of our results, we use MAGMA to provide many examples of new MDS codes over F p m and F p m + uF p m .

Information Theory, IEEE Transactions on, 1999

46 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 1, JANUARY 1999 On Lowest Density MDS Codes Mario Blaum, Senior Member, IEEE, and Ron M. Roth, Senior Member, IEEE Abstract— Let q denote the finite field GF (q) and let b be a positive integer. ...