In distributed storage systems built using commodity hardware, it is necessary to have data redundancy in order to ensure system reliability. In such systems, it is also often desirable to be able to quickly repair storage nodes that fail. We consider a scheme-introduced by El Rouayheb and Ramchandran-which uses combinatorial block design in order to design storage systems that enable efficient (and exact) node repair. In this work, we investigate systems where node sizes may be much larger than replication degrees, and explicitly provide algorithms for constructing these storage designs. Our designs, which are related to projective geometries, are based on the construction of bipartite cage graphs (with girth 6) and the concept of mutually-orthogonal Latin squares. Via these constructions, we can guarantee that the resulting designs require the fewest number of storage nodes for the given parameters, and can further show that these systems can be easily expanded without need for frequent reconfiguration.

In interference alignment, users sharing a wireless channel are each able to achieve data rates of up to half of the non-interfering channel capacity, no matter the number of users. In an ergodic setting, this is achieved by pairing complementary channel realizations in order to amplify signals and cancel interference. However, this scheme has the possibility for large delays in decoding message symbols. We show that delay can be mitigated by using outputs from potentially more than two channel realizations, although data rate may be reduced. We further demonstrate the tradeoff between rate and delay via a time-sharing strategy. Our analysis considers Gaussian channels; an extension to finite field channels is also possible.

Abstract-The non-uniform demand network coding problem is posed as a single-source and multiple-sink network transmission problem where the sinks may have heterogeneous demands. In contrast with multicast problems, non-uniform demand problems are concerned with the amounts of data received by each sink, rather than the specifics of the received data. In this work, we enumerate non-uniform network demand scenarios under which network coding solutions can be found in polynomial time. This is accomplished by relating the demand problem with the graph coloring problem, and then applying results from the strong perfect graph theorem to identify coloring problems which can be solved in polynomial time. This characterization of efficiently-solvable non-uniform demand problems is an important step in understanding such problems, as it allows us to better understand situations under which the NP-complete problem might be tractable.