# Network Coding: An Introduction

Linear network coding may be used to improve a network's throughput, efficiency and scalability , as well as resilience to attacks and eavesdropping. Instead of simply relaying the packets of information they receive, the nodes of a network take several packets and combine them together for transmission. This may be used to attain the maximum possible information flow in a network. It has been mathematically proven in theory that linear coding is enough to achieve the upper bound in multicast problems with one source.

Note that, since operations are computed in a finite field, the generated message is of the same length as the original messages. Sink nodes receive these network coded messages, and collect them in a matrix.

## Network Coding: An Introduction

The original messages can be recovered by performing Gaussian elimination on the matrix. Karl Menger proved that there is always a set of edge-disjoint paths achieving the upper bound in a unicast scenario, known as the max-flow min-cut theorem. Later, the Fordâ€”Fulkerson algorithm was proposed to find such paths in polynomial time. Then, Edmonds proved in the paper "Edge-Disjoint Branchings" the upper bound in the broadcast scenario is also achievable, and proposed a polynomial time algorithm. However, the situation in the multicast scenario is more complicated, and in fact, such an upper bound can't be reached using traditional routing ideas.

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Ahlswede, et al. The butterfly network [2] is often used to illustrate how linear network coding can outperform routing. Two source nodes at the top of the picture have information A and B that must be transmitted to the two destination nodes at the bottom , which each want to know both A and B.

## Network Coding: An Introduction by Tracey Ho

Each edge can carry only a single value we can think of an edge transmitting a bit in each time slot. If only routing were allowed, then the central link would be only able to carry A or B, but not both. Suppose we send A through the center; then the left destination would receive A twice and not know B at all.

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Sending B poses a similar problem for the right destination. We say that routing is insufficient because no routing scheme can transmit both A and B simultaneously to both destinations. Random linear network coding [6] is a simple yet powerful encoding scheme, which in broadcast transmission schemes allows close to optimal throughput using a decentralized algorithm.

Nodes transmit random linear combinations of the packets they receive, with coefficients chosen from a Galois field. If the field size is sufficiently large, the probability that the receiver s will obtain linearly independent combinations and therefore obtain innovative information approaches 1. It should however be noted that, although random linear network coding has excellent throughput performance, if a receiver obtains an insufficient number of packets, it is extremely unlikely that they can recover any of the original packets.

This can be addressed by sending additional random linear combinations until the receiver obtains the appropriate number of packets. Linear network coding is still a relatively new subject. Based on previous studies, there are three important open issues in RLNC:. Enter your mobile number or email address below and we'll send you a link to download the free Kindle App.

Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required. Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? Network coding promises to significantly impact the way communications networks are designed, operated, and understood.

### 1. Introduction

This book presents a unified and intuitive overview of the theory, applications, challenges, and future directions of this emerging field, and is a must-have resource for those working in wireline or wireless networking. This is an essential resource for graduate students and researchers in electronic and computer engineering and for practitioners in the communications industry.

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Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Summary In this chapter we introduce the concept of network coding and how it can be used in a network.