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 highredundancy regime of k/n ≤ 1/2, but the codes have an exponential size (w.r.t. k) for the practically important lowredundancy regime of k/n > 1/2. In this paper, we construct a class of polynomial size codes in this low redundancy regime.
We consider the problem of error control for receiverdriven layered multicast of audio and video over the Internet. The sender injects into the network multiple source layers and multiple channel coding (parity) layers, some of which are delayed relative to the source. Each receiver subscribes to the number of source layers and the number of parity layers that optimizes the receiver's quality for its available bandwidth and packet loss probability. We augment this layered FEC system with layered pseudo-ARQ. Although feedback is normally problematic in broadcast situations, ARQ can be simulated by having the receivers subscribe and unsubscribe to the delayed parity layers to receive missing information. This pseudo-ARQ scheme avoids an implosion of repeat requests at the sender and is scalable to an unlimited number of receivers. We show gains of 4-18 dB on channels with 20% loss over systems without error control and additional gains of 1-13 dB when FEC is augmented by pseudo-ARQ in a hybrid system. Optimal error control in the hybrid system is achieved by an optimal policy for a Markov decision process.
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