Parity Columns Research Articles

Two Classes of Binary MDS Array Codes With Asymptotically Optimal Repair for Any Single Column

An $m\times (k+r)$ binary maximum distance separable (MDS) array code contains $k$ information columns and $r$ parity columns with each entry being a bit, where any $k$ out of $k+r$ columns can recover the $k$ information columns. When there is a failed column, it is critical to minimize the repair bandwidth that is the total number of bits downloaded from $d$ out of $k+r-1$ surviving columns in repairing the failed column. In this article, we first propose two explicit constructions of binary MDS array codes that have asymptotically optimal repair bandwidth for any information column, where $r\geq 2$ and $d=k+r-1$ for the first construction, and $r\geq 4$ is an even number and $d=k+\frac {r}{2}-1$ for the second construction. By applying a generic transformation for the proposed two classes of binary MDS array codes, we then obtain two classes of new binary MDS array codes that also have optimal repair bandwidth for any parity column and asymptotically optimal repair bandwidth for any information column.

A New Design of Binary MDS Array Codes With Asymptotically Weak-Optimal Repair

Binary maximum distance separable (MDS) array codes are a special class of erasure codes for distributed storage that not only provides fault tolerance with minimum storage redundancy but also achieves low computational complexity. They are constructed by encoding $k$ information columns into $r$ parity columns, in which each element in a column is a bit, such that any $k$ out of the $k+r$ columns suffice to recover all information bits. In addition to providing fault tolerance, it is critical to improve repair performance in practical applications. Specifically, if a single column fails, then our goal is to minimize the repair bandwidth by downloading the least amount of bits from $d$ healthy columns, where $k\leq d\leq k+r-1$ . If one column of an MDS code is failed, it is known that we need to download at least $1/(d-k+1)$ fraction of the data stored in each of the $d$ healthy columns. If this lower bound is achieved for the repair of the failure column from accessing arbitrary $d$ healthy columns, we say that the MDS code has optimal repair. However, if such lower bound is only achieved by $d$ specific healthy columns, then we say the MDS code has weak-optimal repair. In this paper, we propose two explicit constructions of binary MDS array codes with more parity columns (i.e., $r\geq 3$ ) that achieve asymptotically weak-optimal repair, where $k+1\leq d\leq k+\lfloor (r-1)/2\rfloor $ , and “asymptotic” means that the repair bandwidth achieves the minimum value asymptotically in $d$ . Codes in the first construction have odd number of parity columns and asymptotically weak-optimal repair for anyone information failure, while codes in the second construction have even number of parity columns and asymptotically weak-optimal repair for any one column failure.

Binary MDS Array Codes With Optimal Repair

Consider a binary maximum distance separable (MDS) array code composed of an $m\times (k+r)$ array of bits with $k$ information columns and $r$ parity columns, such that any $k$ out of $k+r$ columns suffice to reconstruct the $k$ information columns. Our goal is to provide {\em optimal repair access} for binary MDS array codes, meaning that the bandwidth triggered to repair any single failed information or parity column is minimized. In this paper, we propose a generic transformation framework for binary MDS array codes, using EVENODD codes as a motivating example, to support optimal repair access for $k+1\le d \le k+r-1$, where $d$ denotes the number of non-failed columns that are connected for repair; note that when $d<k+r-1$, some of the chosen $d$ columns in repairing a failed column are specific. In addition, we show how our transformation framework applies to an example of binary MDS array codes with asymptotically optimal repair access of any single information column and enables asymptotically or exactly optimal repair access for any column. Furthermore, we present a new transformation for EVENODD codes with two parity columns such that the existing efficient repair property of any information column is preserved and the repair access of parity column is optimal.

A Unified Form of EVENODD and RDP Codes and Their Efficient Decoding

Array codes are used widely in data storage systems such as redundant array of independent disks. The row-diagonal parity (RDP) codes and EVENODD codes are two popular double-parity array codes. The increasing capacity of hard disks demands better fault tolerance by using array codes with three or more parity disks. Although many extensions of RDP and EVENODD codes have been proposed, their main drawback is high decoding complexity. In this paper, we propose a unified form of RDP and EVENODD codes under which RDP codes can be treated as shortened EVENODD codes. Moreover, an efficient decoding algorithm based on an LU factorization of a Vandermonde matrix is proposed. The LU decoding method is applicable to all the erasure patterns of RDP and EVENODD codes with three parity columns. It is also applicable to the erasure decoding of RDP and EVENODD codes with more than three parity columns when the number of continuous surviving parity columns is no less than the number of erased information columns and the first parity column has not failed. The proposed efficient decoding algorithm is also applicable to other Vandermonde array codes, with less decoding complexity than that of the existing method.

A class of binary MDS array codes with asymptotically weak-optimal repair

Binary maximum distance separable (MDS) array codes contain $k$ information columns and $r$ parity columns in which each entry is a bit that can tolerate $r$ arbitrary erasures. When a column in an MDS code fails, it has been proven that we must download at least half of the content from each helper column if $k+1$ columns are selected as the helper columns. If the lower bound is achieved such that the $k+1$ helper columns can be selected from any $k+r-1$ surviving columns, then the repair is an optimal repair. Otherwise, if the lower bound is achieved with $k+1$ specific helper columns, the repair is a weak-optimal repair. This paper proposes a class of binary MDS array codes with $k\geq~3$ and $r\geq~2$ that asymptotically achieve weak-optimal repair of an information column with $k+1$ helper columns. We show that there exist many encoding matrices such that the corresponding binary MDS array codes can asymptotically achieve weak-optimal repair for repairing any information column.

On the MDS Condition of Blaum–Bruck–Vardy Codes With Large Number Parity Columns

Binary maximum distance separable (MDS) array codes are widely used in storage systems. EVENODD code and row-diagonal parity (RDP) code are two well-known binary MDS array codes with two parity columns. The Blaum–Bruck–Vardy (BBV) code is an extension of the double-erasure-correcting EVENODD code. It is known that the BBV code is always MDS for three parity columns, and sufficient conditions for up to eight parity columns to be MDS are also known. However, the MDS condition for more than eight parity columns is an open problem since then. In this letter, we study the MDS condition of BBV code and give a sufficient MDS condition of the BBV code with more than eight parity columns.

MDS array codes with independent parity symbols

A new family of maximum distance separable (MDS) array codes is presented. The code arrays contain p information columns and r independent parity columns, each column consisting of p-1 bits, where p is a prime. We extend a previously known construction for the case r=2 to three and more parity columns. It is shown that when r=3 such extension is possible for any prime p. For larger values of r, we give necessary and sufficient conditions for our codes to be MDS, and then prove that if p belongs to a certain class of primes these conditions are satisfied up to r/spl les/8. One of the advantages of the new codes is that encoding and decoding may be accomplished using simple cyclic shifts and XOR operations on the columns of the code array. We develop efficient decoding procedures for the case of two- and three-column errors. This again extends the previously known results for the case of a single-column error. Another primary advantage of our codes is related to the problem of efficient information updates. We present upper and lower bounds on the average number of parity bits which have to be updated in an MDS code over GF (2/sup m/), following an update in a single information bit. This average number is of importance in many storage applications which require frequent updates of information. We show that the upper bound obtained from our codes is close to the lower bound and, most importantly, does not depend on the size of the code symbols.