Upper and Lower Bounds for the Degree of Groebner Bases
EUROSAM '84 Proceedings of the International Symposium on Symbolic and Algebraic Computation
An algebraic approach to network coding
IEEE/ACM Transactions on Networking (TON)
Graph Theory with Applications to Engineering and Computer Science (Prentice Hall Series in Automatic Computation)
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
Efficient algorithms for solving overdefined systems of multivariate polynomial equations
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Linearity and solvability in multicast networks
IEEE Transactions on Information Theory
Polynomial time algorithms for multicast network code construction
IEEE Transactions on Information Theory
Insufficiency of linear coding in network information flow
IEEE Transactions on Information Theory
Information flow decomposition for network coding
IEEE Transactions on Information Theory
A Random Linear Network Coding Approach to Multicast
IEEE Transactions on Information Theory
Linear Network Codes and Systems of Polynomial Equations
IEEE Transactions on Information Theory
| Hi-index | 754.84 |
In the algebraic view, the solution to a network coding problem is seen as a variety specified by a system of polynomial equations typically derived by using edge-to-edge gains as variables. The output from each sink is equated to its demand to obtain polynomial equations. In this paper, we propose a method to derive the polynomial equations using source-to-sink path gains as the variables. In the path gain formulation, we show that linear and quadratic equations suffice; therffore, network coding becomes equivalent to a system of polynomial equations of maximum degree 2. We present algorithms for generating the equations in the path gains and for converting path gain solutions to edge-to-edge gain solutions. Because of the low degree, simplification is readily possible for the system of equaltions obtained using path gains. Using small-sized network coding: problems, we show that the path gain approach results in simpler equations and determines solvability of the problem in certain cases. On a larger network (with 87 nodes and 161 edges), we show how the path gain approach continues to provide deterministic solutions to some network coding problems.