We consider the delay of network coding compared to routing with retransmissions in packet erasure networks with probabilistic erasures. We investigate the sublinear term in the block delay required for unicasting n packets and show that there is an unbounded gap between network coding and routing. In particular, we show that delay benefit of network coding scales at least as √n. Our analysis of the delay function for the routing strategy involves a major technical challenge of computing the expectation of the maximum of two negative binomial random variables. Previous characterizations of this expectation are approximate; we derive an exact characterization and analyze its scaling behavior, which may be of independent interest. We also use a martingale bounded differences argument to show that the actual coding delay is concentrated around its expectation.

This work considers the multiple-access multicast error-correction scenario over a packetized network withzmalicious edge adversaries. The network has min-cutmand packets of lengthl, and each sink demands all information from the set of sourcesS. The capacity region is characterized for both a “side-channel” model (where sources and sinks share some random bits that are secret from the adversary) and an “omniscient” adversarial model (where no limitations on the adversary's knowledge are assumed). In the “side-channel” adversarial model, the use of a secret channel allows higher rates to be achieved compared to the “omniscient” adversarial model, and a polynomial-complexity capacity-achieving code is provided. For the “omniscient” adversarial model, two capacity-achieving constructions are given: the first is based on random subspace code design and has complexity exponential inlm, while the second uses a novel multiple-field-extension technique and has O(lm|S|) complexity, which is polynomial in the network size. Our code constructions are “end-to-end” in that all nodes except the sources and sinks are oblivious to the adversaries and may simply implement predesigned linear network codes (random or otherwise). Also, the sources act independently without knowledge of the data from other sources.

We investigate the problem of maintaining an encoded distributed storage system when some nodes contain adversarial errors. Using the error-correction capabilities that are built into the existing redundancy of the system, we propose a simple linear hashing scheme to detect errors in the storage nodes. Our main result is that for storing a data object of total size M using an (n, k) MDS code over a finite field Fq, up to t1 = ⌊(n - k)/2⌋ errors can be detected, with probability of failure smaller than 1/M, by communicating only O(n(n-k) logM) bits to a trusted verifier. Our result constructs small projections of the data that preserve the errors with high probability and builds on a pseudorandom generator that fools linear functions. The transmission rate achieved by our scheme is asymptotically equal to the min-cut capacity between the source and any receiver.

We consider the delay of network coding compared to routing for a family of simple networks with parallel links. We investigate the sub-linear term in the block delay required for unicasting n packets and show that there is an unbounded gap between network coding and routing. In particular, we show that delay benefit of network coding is scaling at least as fast as √n. The main technical contribution involves showing that the delay function for the routing retransmission strategy is unbounded. This problem is equivalent to computing the expected maximum of two negative binomial random variables. This problem has also been addressed previously and we derive the first exact characterization which might be of independent interest.

The network communication scenario where one or more receivers request all the information transmitted by different sources is considered. We introduce the first polynomial-time (in network size) network codes that achieve any point inside the rate-region for the problem of multiple-source multicast in the presence of malicious errors, for any fixed number of sources. Our codes are fully distributed and different sources require no knowledge of the data transmitted by their peers. Our codes are “end-to-end”, that is, all nodes apart from the sources and the receivers are oblivious to the adversaries present in the network and simply implement random linear network coding.

This paper presents a new progressive distributed estimation scheme (DES) along with the power scheduling among sensors under AWGN channels. The progressive DES consists of a transmission bit allocation scheme and a quasi best linear unbiased estimate (BLUE) of the unknown parameter at each sensor. This scheme is shown to outperform the traditional progressive DES. Moreover, the power scheduling among sensors, which minimizes the total transmission power subject to the desired MSE performance tolerance, is formulated as a convex problem and the optimal solution is derived analytically.

We analyze a simple network where a source and a receiver are connected by a line of erasure channels of different reliabilities. Recent prior work has shown that random linear network coding can achieve the min-cut capacity and therefore the asymptotic rate is determined by the worst link of the line network. In this paper we investigate the delay for transmitting a batch of packets, which is a function of all the erasure probabilities and the number of packets in the batch. We show a monotonicity result on the delay function and derive simple expressions which characterize the expected delay behavior of line networks. Further, we use a martingale bounded differences argument to show that the actual delay is tightly concentrated around its expectation.

For wireless fading networks where M source nodes wish to multicast their information to N destination nodes via a common relay, it has been shown that network coding at the relay can reduce the number of required transmissions, if the destination nodes overhear M - 1 source nodes transmitting to the relay. Nested coding is a recently proposed alternative where, unlike network coding, physical and network layers are jointly designed. We find that for a given network throughput ideal nested coding in many situations can lead to a significant increase in transmission reliability compared to network coding. We also consider the case of feedback and retransmissions when received packets fail to be decoded. For the case of two source nodes it is shown that by using nested codes the expected number of relay transmissions is reduced. However, for a larger number of source nodes the gain depends on the employed nested coding strategy at the relay.