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An (n, k) maximum distance separable (MDS) code can be used to store data in n storage nodes, such that the system can tolerate the failure of any (n−k) storage nodes. Recently, MDS codes have been constructed which satisfy an additional optimal repair property as follows: the failure of a single storage node can be repaired by downloading a fraction of 1/(n − k) of the data stored in every surviving storage node. In previous constructions satisfying this optimal repair property, the size of the code is polynomial in k for the high-redundancy regime of k/n ≤ 1/2, but the codes have an exponential size (w.r.t. k) for the practically important low-redundancy regime of k/n > 1/2. In this paper, we construct a class of polynomial size codes in this low redundancy regime. [ABSTRACT FROM PUBLISHER]

In this paper, we develop an interference alignment framework for multi-source (non-multicast) network coding applications. The framework developed here is based on using a tensor product structure for the network coding matrices at nodes in a network. The framework is presented in this paper in the specific context of exact repair for distributed storage, which is a multi-source network coding application. Using this framework, we generalize previously known MDS codes with optimal exact repair in distributed storage. [ABSTRACT FROM PUBLISHER]