On Error Decoding of Locally Repairable and Partial MDS Codes

We consider error decoding of locally repairable codes (LRC) and partial MDS (PMDS) codes through interleaved decoding. For a specific class of LRCs we investigate the success probability of interleaved decoding. For PMDS codes we show that there is a wide range of parameters for which interleaved decoding can increase their decoding radius beyond the minimum distance with the probability of successful decoding approaching $1$, when the code length goes to infinity

Optimal Rebuilding of Multiple Erasures in MDS Codes

MDS 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 $r$ node erasures by accessing all the remaining information in the surviving nodes. However, in practice, $e$ erasures is a more likely failure event, for $1\le e<r$. Hence, a natural question is how much information do we need to access in order to rebuild $e$ storage nodes? We define the rebuilding ratio as the fraction of remaining information accessed during the rebuilding of $e$ erasures. In our previous work we constructed MDS codes, called zigzag codes, that achieve the optimal rebuilding ratio of $1/r$ for the rebuilding of any systematic node when $e=1$, however, all the information needs to be accessed for the rebuilding of the parity node erasure. The (normalized) repair bandwidth is defined as the fraction of information transmitted from the remaining nodes during the rebuilding process. For codes that are not necessarily MDS, Dimakis et al. proposed the regenerating codes framework where any $r$ erasures can be corrected by accessing some of the remaining information, and any $e=1$ erasure can be rebuilt from some subsets of surviving nodes with optimal repair bandwidth. In this work, we study 3 questions on rebuilding of codes: (i) We show a fundamental trade-off between the storage size of the node and the repair bandwidth similar to the regenerating codes framework, and show that zigzag codes achieve the optimal rebuilding ratio of $e/r$ for MDS codes, for any $1\le e\le r$. (ii) We construct systematic codes that achieve optimal rebuilding ratio of $1/r$, for any systematic or parity node erasure. (iii) We present error correction algorithms for zigzag codes, and in particular demonstrate how these codes can be corrected beyond their minimum Hamming distances.Comment: There is an overlap of this work with our two previous submissions: Zigzag Codes: MDS Array Codes with Optimal Rebuilding; On Codes for Optimal Rebuilding Access. arXiv admin note: text overlap with arXiv:1112.037

Universal secure rank-metric coding schemes with optimal communication overheads

We study the problem of reducing the communication overhead from a noisy wire-tap channel or storage system where data is encoded as a matrix, when more columns (or their linear combinations) are available. We present its applications to reducing communication overheads in universal secure linear network coding and secure distributed storage with crisscross errors and erasures and in the presence of a wire-tapper. Our main contribution is a method to transform coding schemes based on linear rank-metric codes, with certain properties, to schemes with lower communication overheads. By applying this method to pairs of Gabidulin codes, we obtain coding schemes with optimal information rate with respect to their security and rank error correction capability, and with universally optimal communication overheads, when $n \leq m$, being $n$ and $m$ the number of columns and number of rows, respectively. Moreover, our method can be applied to other families of maximum rank distance codes when $n > m$. The downside of the method is generally expanding the packet length, but some practical instances come at no cost.Comment: 21 pages, LaTeX; parts of this paper have been accepted for presentation at the IEEE International Symposium on Information Theory, Aachen, Germany, June 201

Optimal Locally Repairable and Secure Codes for Distributed Storage Systems

This paper aims to go beyond resilience into the study of security and local-repairability for distributed storage systems (DSS). Security and local-repairability are both important as features of an efficient storage system, and this paper aims to understand the trade-offs between resilience, security, and local-repairability in these systems. In particular, this paper first investigates security in the presence of colluding eavesdroppers, where eavesdroppers are assumed to work together in decoding stored information. Second, the paper focuses on coding schemes that enable optimal local repairs. It further brings these two concepts together, to develop locally repairable coding schemes for DSS that are secure against eavesdroppers. The main results of this paper include: a. An improved bound on the secrecy capacity for minimum storage regenerating codes, b. secure coding schemes that achieve the bound for some special cases, c. a new bound on minimum distance for locally repairable codes, d. code construction for locally repairable codes that attain the minimum distance bound, and e. repair-bandwidth-efficient locally repairable codes with and without security constraints.Comment: Submitted to IEEE Transactions on Information Theor