Itzhak Tamo

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 [1/2] and [3/4]; 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 [1/2]. In general, we construct a new family of r-erasure correcting MDS array codes that has optimal rebuilding ratio of [1/(r)] in the case of a single erasure. Our array codes have efficient encoding and decoding algorithms (for the cases r=2 and r=3, they use a finite field of size 3 and 4, respectively) and an optimal update property.

A Family of Optimal Locally Recoverable Codes

A code over a finite alphabet is called locally recoverable code (LRC code) if every symbol in the encoding is a function of a small number (at most r) other symbols. We present a family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality. The codes can be constructed over a finite field alphabet of any size that exceeds the code length. The codewords are obtained as evaluations of specially constructed polynomials over a finite field, and reduce to Reed-Solomon codes if the locality parameter r is set to be equal to the code dimension. The recovery procedure is performed by polynomial interpolation over r points. We also construct codes with several disjoint recovering sets for every symbol. This construction enables the system to conduct several independent and simultaneous recovery processes of a specific symbol by accessing different parts of the codeword. This property enables high availability of frequently accessed data.

Optimal locally repairable codes and connections to matroid theory

Petabyte-scale distributed storage systems are currently transitioning to erasure codes to achieve higher storage efficiency. Classical codes like Reed-Solomon are highly suboptimal for distributed environments due to their high overhead in single-failure events. Locally Repairable Codes (LRCs) form a new family of codes that are repair efficient. In particular, LRCs minimize the number of nodes participating in single node repairs during which they generate small network traffic. Two large-scale distributed storage systems have already implemented different types of LRCs: Windows Azure Storage and the Hadoop Distributed File System RAID used by Facebook. The fundamental bounds for LRCs, namely the best possible distance for a given code locality, were recently discovered, but few explicit constructions exist. In this work, we present an explicit and simple to implement construction of optimal LRCs, for code parameters previously established by existence results. For the analysis of the optimality of our code, we derive a new result on the matroid represented by the codes generator matrix.

Correcting Limited-Magnitude Errors in the Rank-Modulation Scheme

We study error-correcting codes for permutations under the infinity norm, motivated by a novel storage scheme for flash memories called rank modulation. In this scheme, a set of n flash cells are combined to create a single virtual multilevel cell. Information is stored in the permutation induced by the cell charge levels. Spike errors, which are characterized by a limited-magnitude change in cell charge levels, correspond to a low-distance change under the infinity norm. We define codes protecting against spike errors, called limited-magnitude rank-modulation codes (LMRM codes), and present several constructions for these codes, some resulting in optimal codes. These codes admit simple recursive, and sometimes direct, encoding and decoding procedures. We also provide lower and upper bounds on the maximal size of LMRM codes both in the general case, and in the case where the codes form a subgroup of the symmetric group. In the asymptotic analysis, the codes we construct outperform the Gilbert-Varshamov-like bound estimate.

MDS array codes with optimal rebuilding

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 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 1 over 2 and 3 over 4, 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 2-erasure correcting code, the rebuilding ratio is 1 over 2. In general, we construct a new family of r-erasure correcting MDS array codes that has optimal rebuilding ratio of 1 over r in the case of a single erasure. Our array codes have efficient encoding and decoding algorithms (for the case r = 2 they use a finite field of size 3) and an optimal update property.

On codes for optimal rebuilding access

MDS (maximum distance separable) array codes are widely used in storage systems due to their computationally efficient encoding and decoding procedures. An MDS code with r redundancy nodes can correct any r erasures by accessing (reading) all the remaining information in both the systematic nodes and the parity (redundancy) nodes. However, in practice, a single erasure is the most likely failure event; hence, a natural question is how much information do we need to access in order to rebuild a single storage node? We define the rebuilding ratio as the fraction of remaining information accessed during the rebuilding of a single erasure. In our previous work we constructed array codes that achieve the optimal rebuilding ratio of 1/r for the rebuilding of any systematic node, however, all the information needs to be accessed for the rebuilding of the parity nodes. Namely, constructing array codes with a rebuilding ratio of 1/r for an arbitrary erasure was left as an open problem. In this paper, we solve this open problem and present array codes that achieve the lower bound of 1/r for rebuilding any single systematic or parity node.

Bounds on locally recoverable codes with multiple recovering sets

A locally recoverable code (LRC code) is a code over a finite alphabet such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. Bounds on the rate and distance of such codes have been extensively studied in the literature. In this paper we derive upper bounds on the rate and distance of codes in which every symbol has t ≥ 1 disjoint recovering sets.

Long MDS codes for optimal repair bandwidth

MDS codes are erasure-correcting codes that can correct the maximum number of erasures given the number of redundancy or parity symbols. If an MDS code has r parities and no more than r erasures occur, then by transmitting all the remaining data in the code one can recover the original information. However, it was shown that in order to recover a single symbol erasure, only a fraction of 1/r of the information needs to be transmitted. This fraction is called the repair bandwidth (fraction). Explicit code constructions were given in previous works. If we view each symbol in the code as a vector or a column, then the code forms a 2D array and such codes are especially widely used in storage systems. In this paper, we ask the following question: given the length of the column l, can we construct high-rate MDS array codes with optimal repair bandwidth of 1/r, whose code length is as long as possible? In this paper, we give code constructions such that the code length is (r + l)logr l.

An Improved Sub-Packetization Bound for Minimum Storage Regenerating Codes

Distributed storage systems employ codes to provide resilience to failure of multiple storage disks. In particular, an (n, k) maximum distance separable (MDS) code stores k symbols in n disks such that the overall system is tolerant to a failure of up to n - k disks. However, access to at least k disks is still required to repair a single erasure. To reduce repair bandwidth, array codes are used where the stored symbols or packets are vectors of length ℓ. The MDS array codes have the potential to repair a single erasure using a fraction 1/(n - k) of data stored in the remaining disks. We introduce new methods of analysis, which capitalize on the translation of the storage system problem into a geometric problem on a set of operators and subspaces. In particular, we ask the following question: for a given (n, k), what is the minimum vector-length or subpacketization factor ℓ required to achieve this optimal fraction? For exact recovery of systematic disks in an MDS code of low redundancy, i.e., k/n > 1/2, the best known explicit codes have a subpacketization factor ℓ, which is exponential in k. It has been conjectured that for a fixed number of parity nodes, it is in fact necessary for ℓ to be exponential in k. In this paper, we provide a new log-squared converse bound on k for a given ℓ, and prove that k ≤ 2 log2 I(logδ ℓ + 1), for an arbitrary number of parity nodes r = n - k, where δ = r/(r - 1).

Bounds on the Parameters of Locally Recoverable Codes

A locally recoverable code (LRC code) is a code over a finite alphabet, such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. In this paper, we derive new finite-length and asymptotic bounds on the parameters of LRC codes. For LRC codes with a single recovering set for every coordinate, we derive an asymptotic Gilbert-Varshamov type bound for LRC codes and find the maximum attainable relative distance of asymptotically good LRC codes. Similar results are established for LRC codes with two disjoint recovering sets for every coordinate. For the case of multiple recovering sets (the availability problem), we derive a lower bound on the parameters using expander graph arguments. Finally, we also derive finite-length upper bounds on the rate and the distance of LRC codes with multiple recovering sets.