Network Coding Capacity Regions via Entropy Functions

• Published in: IEEE transactions on information theory Vol. 60; no. 9; pp. 5347 - 5374
• Main Authors: Chan, Terence H., Grant, Alex
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.09.2014; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Coding, codes; Decoding; Encoding; Entropy; Error correction & detection; Error probability; Exact sciences and technology; Indexes; Information theory; Information, signal and communications theory; Linear codes; Network coding; Random variables; Routing; Secret; Signal and communications theory; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Transmission and modulation (techniques and equipments); Network coding; achievable rate regions; linearity constraint; incremental multicast; Packet switching; Source coding; Error rate; Routing; Entropy; Decoding; Data broadcast; Information dissemination; Multicast; Linear code; Linear coding; Linearity; Optimal code; Secrecy
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2014.2334291

In this paper, we use entropy functions to characterize the set of rate-capacity tuples achievable with either zero decoding error, or vanishing decoding error, for general network coding problems for acyclic networks. We show that when sources are colocated, the outer bound is tight and the sets of zero-error achievable and vanishing-error achievable rate-capacity tuples are the same. Then, we extend this paper to networks subject to linear encoding constraints, routing constraints (where some or all nodes can only perform routing), and secrecy constraints. Finally, we show that even for apparently simple networks, design of optimal codes may be difficult. In particular, we prove that for the incremental multicast problem and for the single-source secure network coding problem, characterization of the achievable set can be very hard and linear network codes may not be optimal.