Partial MDS (PMDS) and sector-disk (SD) codes are classes of erasure codes that combine locality with strong erasure correction capabilities. We construct PMDS and SD codes where each local code is a bandwidth-optimal regenerating MDS code. The constructions require significantly smaller field size than the only other construction known in literature.
Guo, Kopparty and Sudan have initiated the study of error-correcting codes derived by lifting of affine-invariant codes. Lifted Reed-Solomon (RS) codes are defined as the evaluation of polynomials in a vector space over a field by requiring their restriction to every line in the space to be a codeword of the RS code. In this paper, we investigate lifted RS codes and discuss their application to batch codes, a notion introduced in the context of private information retrieval and load-balancing in distributed storage systems. First, we improve the estimate of the code rate of lifted RS codes for lifting parameter $m\ge 3$ and large field size. Second, a new explicit construction of batch codes utilizing lifted RS codes is proposed. For some parameter regimes, our codes have a better trade-off between parameters than previously known batch codes.
Distributed databases often suffer unequal distribution of data among storage nodes, which is known as `data skew'. Data skew arises from a number of causes such as removal of existing storage nodes and addition of new empty nodes to the database. Data skew leads to performance degradations and thus necessitates `rebalancing' at regular intervals to reduce the amount of skew. We define an $r$-balanced distributed database as a distributed database in which the storage across the nodes has uniform size, and each bit of the data is replicated in $r$ distinct storage nodes. We consider the problem of designing such balanced databases along with associated rebalancing schemes which maintain the $r$-balanced property under node removal and addition operations. We present a class of $r$-balanced databases (parameterized by the number of storage nodes) which have the property of structural invariance, i.e., the databases designed for different number of storage nodes have the same structure. For this class of $r$-balanced databases, we present rebalancing schemes which use coded transmissions between storage nodes, and characterize their communication loads under node addition and removal. We show that the communication cost incurred to rebalance our distributed database for node addition and removal is optimal, i.e., it achieves the minimum possible cost among all possible balanced distributed databases and rebalancing schemes.
We propose a novel technique for constructing a graph representation of a code through which we establish a significant connection between the service rate problem and the well-known fractional matching problem. Using this connection, we show that the service capacity of a coded storage system equals the fractional matching number in the graph representation of the code, and thus is lower bounded and upper bounded by the matching number and the vertex cover number, respectively. This is of great interest because if the graph representation of a code is bipartite, then the derived upper and lower bounds are equal, and we obtain the capacity. Leveraging this result, we characterize the service capacity of the binary simplex code whose graph representation is bipartite. Moreover, we show that the service rate problem can be viewed as a generalization of the multiset primitive batch codes problem.
The secure exact-repair regenerating codes are studied, for distributed storage systems with parameters $(n,k=d,d,\ell)$. The secrecy constraint guarantees that the message remains secure against an eavesdropper who can observe the incoming repair data from all possible nodes to a fixed but unknown subset of (up to) $\ell$ compromised nodes (type II secrecy). A class of secure determinant codes are introduced for all system parameters, and an achievable secrecy trade-off between the per-node storage capacity and repair bandwidth is characterized.