On the capacity of information networks
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Unachievability of network coding capacity
IEEE/ACM Transactions on Networking (TON) - Special issue on networking and information theory
On average throughput and alphabet size in network coding
IEEE/ACM Transactions on Networking (TON) - Special issue on networking and information theory
Minimum-cost multicast over coded packet networks
IEEE/ACM Transactions on Networking (TON) - Special issue on networking and information theory
Network coding theory part II: multiple source
Communications and Information Theory
Network coding theory: single sources
Communications and Information Theory
Foundations and Trends® in Networking
XORs in the air: practical wireless network coding
IEEE/ACM Transactions on Networking (TON)
Capacity regions for multiple unicast flows using inter-session network coding
WICON '07 Proceedings of the 3rd international conference on Wireless internet
Average throughput with linear network coding over finite fields: the combination network case
EURASIP Journal on Wireless Communications and Networking - Advances in Error Control Coding Techniques
RANC: Relay-aided network coding in multi-hop wireless networks
Computer Communications
An overview of network coding for dynamically changing networks
International Journal of Autonomous and Adaptive Communications Systems
General Scheme for Perfect Quantum Network Coding with Free Classical Communication
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Modeling throughput gain of network coding in multi-channel multi-radio wireless ad hoc networks
IEEE Journal on Selected Areas in Communications - Special issue on network coding for wireless communication networks
Cross-layer optimization for wireless multihop networks with pairwise intersession network coding
IEEE Journal on Selected Areas in Communications - Special issue on network coding for wireless communication networks
Wireless network coding in slotted ALOHA with two-hop unbalanced traffic
IEEE Journal on Selected Areas in Communications - Special issue on network coding for wireless communication networks
Improving the multicommodity flow rates with network codes for two sources
IEEE Journal on Selected Areas in Communications - Special issue on network coding for wireless communication networks
A new construction method for networks from matroids
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 4
Feasible alphabets for communicating the sum of sources over a network
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 2
Network coding capacity: a functional dependence bound
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 1
Network coding is highly non-approximable
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
Low-complexity non-uniform demand multicast network coding problems
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
Violating the Ingleton inequality with finite groups
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
Joint scheduling and instantaneously decodable network coding
GLOBECOM'09 Proceedings of the 28th IEEE conference on Global telecommunications
On reliable transmission by adaptive network coding in wireless sensor networks
ICC'09 Proceedings of the 2009 IEEE international conference on Communications
Computing along routes via gossiping
IEEE Transactions on Signal Processing
On the index coding problem and its relation to network coding and matroid theory
IEEE Transactions on Information Theory
Pairwise intersession network coding on directed networks
IEEE Transactions on Information Theory
Path gain algebraic formulation for the scalar linear network coding problem
IEEE Transactions on Information Theory
Joint power allocation for multicast systems with physical-layer network coding
EURASIP Journal on Wireless Communications and Networking - Special issue on physical-layer network coding for wireless cooperative networks
Rate control with pairwise intersession network coding
IEEE/ACM Transactions on Networking (TON)
Network coding-aware routing in wireless networks
IEEE/ACM Transactions on Networking (TON)
Cross-layer optimization of wireless multihop networks with one-hop two-way network coding
Computer Networks: The International Journal of Computer and Telecommunications Networking
Review: Survey of network coding-aware routing protocols in wireless networks
Journal of Network and Computer Applications
Distributed network coding-based opportunistic routing for multicast
Proceedings of the thirteenth ACM international symposium on Mobile Ad Hoc Networking and Computing
On the Multiple-Unicast Capacity of 3-Source, 3-Terminal Directed Acyclic Networks
IEEE/ACM Transactions on Networking (TON)
Hi-index | 755.02 |
It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite-field alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solution over any finite field. However, each example has a linear solution for some vector dimension greater than one. It has been conjectured that every solvable network has a linear solution over some finite-field alphabet and some vector dimension. We provide a counterexample to this conjecture. We also show that if a network has no linear solution over any finite field, then it has no linear solution over any finite commutative ring with identity. Our counterexample network has no linear solution even in the more general algebraic context of modules, which includes as special cases all finite rings and Abelian groups. Furthermore, we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field (exactly 10% greater), so the network is not even asymptotically linearly solvable. It follows that, even for more general versions of linearity such as convolutional coding, filter-bank coding, or linear time sharing, the network has no linear solution.