Efficient network code design for cyclic networks

IEEE Transactions on Information Theory, 2005

The famous max-flow min-cut theorem states that a source node can send information through a network ( ) to a sink node at a rate determined by the min-cut separating and . Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re-encode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures.

This paper proposes Fulcrum network codes, a network coding framework that achieves three seemingly conflicting objectives: (i) to reduce the overhead per coded packet to almost 1 bit per original packet; (ii) to operate the network using only GF operations at intermediate nodes if necessary, dramatically reducing complexity in the network; (iii) to deliver an end-to-end performance that is close Receivers Flow 2 Receivers Flow 1 Network LTE 3G WiFi : Fulcrum network codes allow sources and receivers to operate at higher field sizes to achieve high performance but maintaining compatibility with the GF (2)-only network. Receivers can choose to trade-off delay with decoding effort by choosing to decode with GF (2) or in higher fields.

In forward error correction (FEC) systems, basically two mechanisms namely block coding and convolutional coding are used for error-control coding. The error-control coding processes involve encoding and decoding information symbols to get rid of errors produced by noise in communication channels. In a binary block encoding technique, message words are arranged in blocks of k bits which are converted into code words of n bits, n> k , by adding redundancy bits. Encoding is a compulsory part of error correcting codes and without knowledge of encoding structure the decoding of code words is impossible. In this paper a very simple and most effective technique for systematic encoding of linear binary cyclic codes is used which improves the encoding speed by reducing the computational complexity. In addition to encoding speed, the encoding can be done by hand using XOR (modulo-2) operations without involving computers after generating an encoding table for the code.

2006

In the multicast network coding problem, a source needs to deliver packets to a set of terminals over an underlying communication network . The nodes of the multicast network can be broadly categorized into two groups. The first group incudes encoding nodes, i.e., nodes that generate new packets by combining data received from two or more incoming links. The second group includes forwarding nodes that can only duplicate and forward the incoming packets. Encoding nodes are, in general, more expensive due to the need to equip them with encoding capabilities. In addition, encoding nodes incur delay and increase the overall complexity of the network. Accordingly, in this paper, we study the design of multicast coding networks with a limited number of encoding nodes. We prove that in a directed acyclic coding network, the number of encoding nodes required to achieve the capacity of the network is bounded by 3 2 . Namely, we present (efficiently constructible) network codes that achieve capacity in which the total number of encoding nodes is independent of the size of the network and is bounded by 3 2 . We show that the number of encoding nodes may depend both on and by presenting acyclic coding networks that require ( 2 ) encoding nodes. In the general case of coding networks with cycles, we show that the number of encoding nodes is limited by the size of the minimum feedback link set, i.e., the minimum number of links that must be removed from the network in order to eliminate cycles. We prove that the number of encoding nodes is bounded by (2 + 1) 3 2 , where is the minimum size of a feedback link set. Finally, we observe that determining or even crudely approximating the minimum number of required encoding nodes is an -hard problem.

2012 Conference Record of the Forty Sixth Asilomar Conference on Signals, Systems and Computers (ASILOMAR), 2012

In this paper, we propose a novel opportunistic decoding scheme for network coding decoder which significantly reduces the decoder complexity and increases the throughput. Network coding was proposed to improve the network throughput and reliability, especially for multicast transmissions. Although network coding increases the network performance, the complexity of the network coding decoder algorithm is still high, especially for higher dimensional finite fields or larger network codes. Different software and hardware approaches were proposed to accelerate the decoding algorithm, but the decoder remains to be the bottleneck for high speed data transmission. We propose a novel decoding scheme which exploits the structure of the network coding matrix to reduce the network decoder complexity and improve throughput. We also implemented the proposed scheme on Virtex 7 FPGA and compared our implementation to the widely used Gaussian elimination.

International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings., 2004

IEEE Transactions on Information Theory, 2000

Consider a communication network in which certain source nodes multicast information to other nodes on the network in the multihop fashion where every node can pass on any of its received data to others. We are interested in how fast each node can receive the complete information, or equivalently, what the information rate arriving at each node is. Allowing a node to encode its received data before passing it on, the question involves optimization of the multicast mechanisms at the nodes. Among the simplest coding schemes is linear coding, which regards a block of data as a vector over a certain base field and allows a node to apply a linear transformation to a vector before passing it on. We formulate this multicast problem and prove that linear coding suffices to achieve the optimum, which is the max-flow from the source to each receiving node.

IEEE Transactions on Information Theory, 2000

A condition governing the possibility and impossibility of linear independence among the global encoding kernels of a linear network code is found. Based on this condition, we propose several alternative definitions of generic network codes, which give interpretations of such codes from different perspectives. We also present a unified framework for specifying and constructing different classes of linear network codes. Finally, using the insight obtained from the unified framework, we show that the proofs of some existing results regarding generic network codes can be greatly simplified.

arXiv (Cornell University), 2016

Random linear network coding (RLNC) in theory achieves the max-flow capacity of multicast networks, at the cost of high decoding complexity. To improve the performance-complexity tradeoff, we consider the design of sparse network codes. A generation-based strategy is employed in which source packets are grouped into overlapping subsets called generations. RLNC is performed only amongst packets belonging to the same generation throughout the network so that sparseness can be maintained. In this paper, generation-based network codes with low reception overheads and decoding costs are designed for transmitting of the order of 10 2-10 3 source packets. A low-complexity overhead-optimized decoder is proposed that exploits "overlaps" between generations. The sparseness of the codes is exploited through local processing and multiple rounds of pivoting of the decoding matrix. To demonstrate the efficacy of our approach, codes comprising a binary precode, random overlapping generations, and binary RLNC are designed. The results show that our designs can achieve negligible code overheads at low decoding costs, and outperform existing network codes that use the generation based strategy.

2011 Wireless Advanced, 2011

Adopting a cross-layer approach, in this paper we propose an algorithm for joint routing and network coding. The proposed algorithm jointly assigns routes and designs linear network codes over finite fields to achieve the capacity of the network. The algorithm has a dynamic programming approach where a cost function is used to assign weights to all edges in the network. The cheapest flow is chosen subject to certain encoding constraints in order to achieve the network capacity with network coding while minimizing the network complexity. The effectiveness of the algorithm is demonstrated through carefully chosen examples. We show that the constraints imposed by the joint routing and coding algorithm are necessary for successful decoding at the sinks, and their violation can lead to a failure in achieving the network capacity or an increase in the number of encoding nodes.