Network Codes with Overlapping Chunks over Line Networks: A Case for Linear-Time Codes

In this paper, the problem of designing network codes that are both communicationally and computationally efficient over packet line networks with worst-case schedules is considered. In this context, random linear network codes (dense codes) are asymptotically capacity-achieving, but require highly complex coding operations. To reduce the coding complexity, Maymounkov et al. proposed chunked codes (CC). Chunked codes operate by splitting the message (a collection of packets) into non-overlapping chunks and send a randomly chosen chunk at each transmission time by a dense code. The complexity, that is linear in the chunk size, is thus reduced compared to dense codes. In this paper, the existing analysis of CC is revised, and tighter bounds on the performance of CC are derived. As a result, we prove that (i) CC with sufficiently large chunks are asymptotically capacity-achieving, but with a slower speed of convergence compared to dense codes; and (ii) CC with relatively smaller chunks approach the capacity with an arbitrarily small but non-zero constant gap. To improve the speed of convergence of CC, while maintaining their advantage in reducing the computational complexity, we propose and analyze a new CC scheme with overlapping chunks, referred to as overlapped chunked codes (OCC). We prove that for smaller chunks, which are advantageous due to lower computational complexity, OCC with larger overlaps provide a better tradeoff between the speed of convergence and the message or packet error rate. This implies that for smaller chunks, and with the same computational complexity, OCC outperform CC in terms of the speed of approaching the capacity for sufficiently small target error rate. In fact, we design linear-time OCC with very small chunks (constant in the message size) that are both computationally and communicationally efficient, and that outperform linear-time CC. Finite-length simulation results consistent with the asymptotic analytical results are also presented. Both the analytical and simulation results suggest great potential for the application of OCC for multimedia transmission over packet networks. A preliminary version of this work was partly presented in ITW 2010, Cairo, Egypt, January 2010, and has been accepted in part for presentation at ISIT 2011, Saint-Petersburg, Russia, August 2011.