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 ≤ 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^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

Shortening array codes and the perfect 1-factorization conjecture

The existence of a perfect 1-factorization of the complete graph with n nodes, namely, K_n , for arbitrary even number n, is a 40-year-old open problem in graph theory. So far, two infinite families of perfect 1-factorizations have been shown to exist, namely, the factorizations of K_(p+1) and K_2p , where p is an arbitrary prime number (p > 2) . It was shown in previous work that finding a perfect 1-factorization of K_n is related to a problem in coding, specifically, it can be reduced to constructing an MDS (Minimum Distance Separable), lowest density array code. In this paper, a new method for shortening arbitrary array codes is introduced. It is then used to derive the K_(p+1) family of perfect 1-factorization from the K_2p family. Namely, techniques from coding theory are used to prove a new result in graph theory-that the two factorization families are related

X-code: MDS array codes with optimal encoding

We present a new class of MDS (maximum distance separable) array codes of size n×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

Cyclic Low-Density MDS Array Codes

We construct two infinite families of low density MDS array codes which are also cyclic. One of these families includes the first such sub-family with redundancy parameter r > 2. The two constructions have different algebraic formulations, though they both have the same indirect structure. First MDS codes that are not cyclic are constructed and then by applying a certain mapping to their parity check matrices, non-equivalent cyclic codes with the same distance and density properties are obtained. Using the same proof techniques, a third infinite family of quasi-cyclic codes can be constructed

Low-density MDS codes and factors of complete graphs

We present a class of array code of size n×l, where l=2n or 2n+1, called B-Code. The distances of the B-Code and its dual are 3 and l-1, respectively. The B-Code and its dual are optimal in the sense that i) they are maximum-distance separable (MDS), ii) they have an optimal encoding property, i.e., the number of the parity bits that are affected by change of a single information bit is minimal, and iii) they have optimal length. Using a new graph description of the codes, we prove an equivalence relation between the construction of the B-Code (or its dual) and a combinatorial problem known as perfect one-factorization of complete graphs, thus obtaining constructions of two families of the B-Code and its dual, one of which is new. Efficient decoding algorithms are also given, both for erasure correcting and for error correcting. The existence of perfect one-factorizations for every complete graph with an even number of nodes is a 35 years long conjecture in graph theory. The construction of B-Codes of arbitrary odd length will provide an affirmative answer to the conjecture

Access vs. Bandwidth in Codes for Storage

Maximum distance separable (MDS) codes are widely used in storage systems to protect against disk (node) failures. A node is said to have capacity $l$ over some field $\mathbb{F}$, if it can store that amount of symbols of the field. An $(n,k,l)$ MDS code uses $n$ nodes of capacity $l$ to store $k$ information nodes. The MDS property guarantees the resiliency to any $n-k$ node failures. An \emph{optimal bandwidth} (resp. \emph{optimal access}) MDS code communicates (resp. accesses) the minimum amount of data during the repair process of a single failed node. It was shown that this amount equals a fraction of $1/(n-k)$ of data stored in each node. In previous optimal bandwidth constructions, $l$ scaled polynomially with $k$ in codes with asymptotic rate $<1$. Moreover, in constructions with a constant number of parities, i.e. rate approaches 1, $l$ is scaled exponentially w.r.t. $k$. In this paper, we focus on the later case of constant number of parities $n-k=r$, and ask the following question: Given the capacity of a node $l$ what is the largest number of information disks $k$ in an optimal bandwidth (resp. access) $(k+r,k,l)$ MDS code. We give an upper bound for the general case, and two tight bounds in the special cases of two important families of codes. Moreover, the bounds show that in some cases optimal-bandwidth code has larger $k$ than optimal-access code, and therefore these two measures are not equivalent.Comment: This paper was presented in part at the IEEE International Symposium on Information Theory (ISIT 2012). submitted to IEEE transactions on information theor

Rebuilding for Array Codes in Distributed Storage Systems

In distributed storage systems that use coding, the issue of minimizing the communication required to rebuild a storage node after a failure arises. We consider the problem of repairing an erased node in a distributed storage system that uses an EVENODD code. EVENODD codes are maximum distance separable (MDS) array codes that are used to protect against erasures, and only require XOR operations for encoding and decoding. We show that when there are two redundancy nodes, to rebuild one erased systematic node, only 3/4 of the information needs to be transmitted. Interestingly, in many cases, the required disk I/O is also minimized