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In this paper, the problem of designing network codes that are bothcommunicationally and computationally efficient over packet line networks withworst-case schedules is considered. In this context, random linear networkcodes (dense codes) are asymptotically capacity-achieving, but require highlycomplex coding operations. To reduce the coding complexity, Maymounkov et al.proposed chunked codes (CC). Chunked codes operate by splitting the messageinto non-overlapping chunks and send a randomly chosen chunk at eachtransmission time by a dense code. The complexity, that is linear in the chunksize, is thus reduced compared to dense codes. In this paper, the existinganalysis of CC is revised, and tighter bounds on the performance of CC arederived. As a result, we prove that (i) CC with sufficiently large chunks areasymptotically capacity-achieving, but with a slower speed of convergencecompared to dense codes; and (ii) CC with relatively smaller chunks approachthe capacity with an arbitrarily small but non-zero constant gap. To improvethe speed of convergence of CC, while maintaining their advantage in reducingthe computational complexity, we propose and analyze a new CC scheme withoverlapping chunks, referred to as overlapped chunked codes (OCC). We provethat for smaller chunks, which are advantageous due to lower computationalcomplexity, OCC with larger overlaps provide a better tradeoff between thespeed of convergence and the message or packet error rate. This implies thatfor smaller chunks, and with the same computational complexity, OCC outperformCC in terms of the speed of approaching the capacity for sufficiently smalltarget error rate. In fact, we design linear-time OCC with very small chunks(constant in the message size) that are both computationally andcommunicationally efficient, and that outperform linear-time CC.

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