#### Constant Delay and Constant Feedback Moving Window Network Coding for Wireless Multicast: Design and Asymptotic Analysis

##### Fei Wu, Yin Sun, Yang Yang, Kannan Srinivasan, Ness B. Shroff

A major challenge of wireless multicast is to be able to support a large number of users while simultaneously maintaining low delay and low feedback overhead. In this paper, we develop a joint coding and feedback scheme named Moving Window Network Coding with Anonymous Feedback (MWNC-AF) that successfully addresses this challenge. In particular, we show that our scheme simultaneously achieves both a constant decoding delay and a constant feedback overhead, irrespective of the number of receivers $n$, without sacrificing either throughput or reliability. We explicitly characterize the asymptotic decay rate of the tail of the delay distribution, and prove that transmitting a fixed amount of information bits into the MWNC-AF encoder buffer in each time-slot (called "constant data injection process") achieves the fastest decay rate, thus showing how to obtain delay optimality in a large deviation sense. We then investigate the average decoding delay of MWNC-AF, and show that when the traffic load approaches the capacity, the average decoding delay under the constant injection process is at most one half of that under a Bernoulli injection process. In addition, we prove that the per-packet encoding and decoding complexity of MWNC-AF both scale as $O(\log n)$, with the number of receivers $n$. Our simulations further underscore the performance of our scheme through comparisons with other schemes and show that the delay, encoding and decoding complexity are low even for a large number of receivers, demonstrating the efficiency, scalability, and ease of implementability of MWNC-AF.

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