Natalia Silberstein

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 (DSSs). 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 the stored information. Second, this 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: 1) an improved bound on the secrecy capacity for minimum storage regenerating codes; 2) secure coding schemes that achieve the bound for some special cases; 3) a new bound on minimum distance for locally repairable codes; 4) code construction for locally repairable codes that attain the minimum distance bound; and 5) repair-bandwidth-efficient locally repairable codes with and without security constraints.

Error-Correcting Codes in Projective Spaces Via Rank-Metric Codes and Ferrers Diagrams

Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced row echelon form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the projective space. In this paper, we propose a method to design error-correcting codes in the projective space. We use a multilevel approach to design our codes. First, we select a constant-weight code. Each codeword defines a skeleton of a basis for a subspace in reduced row echelon form. This skeleton contains a Ferrers diagram on which we design a rank-metric code. Each such rank-metric code is lifted to a constant-dimension code. The union of these codes is our final constant-dimension code. In particular, the codes constructed recently by Koetter and Kschischang are a subset of our codes. The rank-metric codes used for this construction form a new class of rank-metric codes. We present a decoding algorithm to the constructed codes in the projective space. The efficiency of the decoding depends on the efficiency of the decoding for the constant-weight codes and the rank-metric codes. Finally, we use puncturing on our final constant-dimension codes to obtain large codes in the projective space which are not constant-dimension.

Optimal locally repairable codes via rank-metric codes

This paper presents a new explicit construction for locally repairable codes (LRCs) for distributed storage systems which possess all-symbol locality and the largest possible minimum distance, or equivalently, can tolerate the maximum number of node failures. This construction, based on maximum rank distance (MRD) Gabidulin codes, provides new optimal vector and scalar LRCs. In addition, the paper also discusses mechanisms by which codes obtained using this construction can be used to construct LRCs with efficient local repair of failed nodes by combination of LRCs with regenerating codes.

Error resilience in distributed storage via rank-metric codes

This paper presents a novel coding scheme for distributed storage systems containing nodes with adversarial errors. The key challenge in such systems is the propagation of erroneous data from a single corrupted node to the rest of the system during a node repair process. This paper presents a concatenated coding scheme which is based on two types of codes: maximum rank distance (MRD) code as an outer code and optimal repair maximal distance separable (MDS) array code as an inner code. Given this, two different types of adversarial errors are considered: the first type considers an adversary that can replace the content of an affected node only once; while the second attack-type considers an adversary that can pollute data an unbounded number of times. This paper proves that the proposed coding scheme attains an upper bound on resilience capacity for the first type of error. Further, the paper presents mechanisms that combine this code with subspace signatures to achieve error resilience for the second type of errors. Finally, the paper concludes by presenting a construction based on MRD codes for optimal locally repairable scalar codes that can tolerate adversarial errors.

Explicit MBR all-symbol locality codes

Node failures are inevitable in distributed storage systems (DSS). To enable efficient repair when faced with such failures, two main techniques are known: Regenerating codes, i.e., codes that minimize the total repair bandwidth; and codes with locality, which minimize the number of nodes participating in the repair process. This paper focuses on regenerating codes with locality, using pre-coding based on Gabidulin codes, and presents constructions that utilize minimum bandwidth regenerating (MBR) local codes. The constructions achieve maximum resilience (i.e., optimal minimum distance) and have maximum capacity (i.e., maximum rate). Finally, the same pre-coding mechanism can be combined with a subclass of fractional-repetition codes to enable maximum resilience and repair-by-transfer simultaneously.

Optimal binary locally repairable codes via anticodes

This paper presents a construction for several families of optimal binary locally repairable codes (LRCs) with small locality (2 and 3). This construction is based on various anticodes. It provides binary LRCs which attain the Cadambe-Mazumdar bound. Moreover, most of these codes are optimal with respect to the Griesmer bound.

Large constant dimension codes and lexicodes

Constant dimension codes, with a prescribed minimum distance, have found recently an application in network coding. All the codewords in such a code are subspaces of

Enumerative Coding for Grassmannian Space

\F_q^n

Subspace Codes Based on Graph Matchings, Ferrers Diagrams, and Pending Blocks

with a given dimension. A computer search for large constant dimension codes is usually inefficient since the search space domain is extremely large. Even so, we found that some constant dimension lexicodes are larger than other known codes. We show how to make the computer search more efficient. In this context we present a formula for the computation of the distance between two subspaces, not necessarily of the same dimension.

Optimal Fractional Repetition Codes Based on Graphs and Designs

The Grassmannian space Gq(n, k) is the set of all k-dimensional subspaces of the vector space Fqn. Recently, codes in the Grassmannian have found an application in network coding. The main goal of this paper is to present efficient enumerative encoding and decoding techniques for the Grassmannian. These coding techniques are based on two different orders for the Grassmannian induced by different representations of k-dimensional subspaces of Fqn. One enumerative coding method is based on a Ferrers diagram representation and on an order for Gq(n, k) based on this representation. The complexity of this enumerative coding is O(k5/2(n - k)5/2) digit operations. Another order of the Grassmannian is based on a combination of an identifying vector and a reduced row echelon form representation of subspaces. The complexity of the enumerative coding, based on this order, is O(nk(n - k) log n log log n) digit operations. A combination of the two methods reduces the complexity on average by a constant factor.