Maximum Distance Separable Array Codes Research Articles

Zigzag Codes: MDS Array Codes With Optimal Rebuilding

Maximum distance separable (MDS) array codes are widely used in storage systems to protect data against erasures. We address the rebuilding ratio problem, namely, in the case of erasures, what is the fraction of the remaining information that needs to be accessed in order to rebuild exactly the lost information? It is clear that when the number of erasures equals the maximum number of erasures that an MDS code can correct, then the rebuilding ratio is 1 (access all the remaining information). However, the interesting and more practical case is when the number of erasures is smaller than the erasure correcting capability of the code. For example, consider an MDS code that can correct two erasures: What is the smallest amount of information that one needs to access in order to correct a single erasure? Previous work showed that the rebuilding ratio is bounded between <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{1} \over {2}}$</tex></formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{3} \over {4}}$</tex></formula> ; however, the exact value was left as an open problem. In this paper, we solve this open problem and prove that for the case of a single erasure with a two-erasure correcting code, the rebuilding ratio is <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{1} \over {2}}$</tex></formula> . In general, we construct a new family of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$r$</tex></formula> -erasure correcting MDS array codes that has optimal rebuilding ratio of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{1} \over {r}}$</tex></formula> in the case of a single erasure. Our array codes have efficient encoding and decoding algorithms (for the cases <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$r=2$</tex> </formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$r=3$</tex></formula> , they use a finite field of size 3 and 4, respectively) and an optimal update property.

Cycle codes of graphs and MDS array codes

Abstract We investigate how to colour edges of graph so that any two colours make up a spanning tree. This problem is equivalent to transforming the cycle code of a graph into a Maximum Distance Separable (MDS) array code. Adopting this graph-theoretical interpretation allows us to provide a compact description of some families of low density MDS array codes in terms of cycle codes of coloured graphs. This includes a short description of Xu et al.'s “B-code”, together with its erasure and error decoding algorithms. We also give a partial answer to Xu et al.'s question about efficient error decoding for the dual B-code. We give alternative families of MDS array cycle codes, and in passing prove that an optimal MDS array cycle code of minimum column distance 4 is given by an appropriate colouring of the Petersen graph. We introduce double array colourings which allow the decoding of column or row errors and provides a new interesting graph theoretical problem. We give infinite families of graphs which admit double array colourings.

New Efficient MDS Array Codes for RAID Part II: Rabin-Like Codes for Tolerating Multiple (greater than or equal to 4) Disk Failures

For pt.1 see ibid., vol.54, no.9, p.1071-1080 (2005). A new class of binary maximum distance separable (MDS) array codes which are based on circular permutation matrices are introduced in this paper. These array codes are used for tolerating multiple (/spl ges/ 4) disk failures in redundant arrays of inexpensive disks (RAID) architecture. The size of the information part is m /spl times/ n, where n is the number of information disks and (m + 1) is a prime integer; the size of the parity-check part is m /spl times/ r, the minimum distance is r + 1, and the number of parity-check disks is r. In practical applications, m can be very large and n ranges from 20 to 50. The code rate is R = n/(n+r). These codes can be used for tolerating up to r disk failures, with very fast encoding and decoding. The complexities of encoding and decoding algorithms are O(rmn) and O(m/sup 3/r/sup 4/), respectively. When r = 4, there need to be 9mn XOR operations for encoding and (9n + 95)(m + 1) XOR operations for decoding.

X-code: MDS array codes with optimal encoding

We present a new class of MDS (maximum distance separable) array codes of size n/spl times/n (n a prime number) called X-code. The X-codes are of minimum column distance 3, namely, they can correct either one column error or two column erasures. The key novelty in X-code is that it has a simple geometrical construction which achieves encoding/update optimal complexity, i.e., a change of any single information bit affects exactly two parity bits. The key idea in our constructions is that all parity symbols are placed in rows rather than 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.