Shuo-Yen Robert Li

Linear network coding

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.

Network coding theory: single sources

Store-and-forward had been the predominant technique for transmitting information through a network until its optimality was refuted by network coding theory. Network coding offers a new paradigm for network communications and has generated abundant research interest in information and coding theory, networking, switching, wireless communications, cryptography, computer science, operations research, and matrix theory.

On the Structure of t-Designs

It is possible to view the combinatorial structures known as (integral) t-designs as

The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain

\mathbb{Z}

Linear Network Coding: Theory and Algorithms

-modules in a natural way. In this note we introduce a polynomial associated to each such

On Convolutional Network Coding

\mathbb{Z}

Network Coding Theory Via Commutative Algebra

-module. Using this association, we quickly derive explicit bases for the important class of submodules which correspond to the so-called null-designs.

N-person Nim andn-person Moore's Games

A martingale argument is used to derive the generating function of the number of i.i.d. experiments it takes to observe a given string of outcomes for the first time. Then, a more general problem can be studied: How many trials does it take to observe a member of a finite set of strings for the first time? It is shown how the answer can be obtained within the framework of hitting times in a Markov chain. For these, a result of independent interest is derived.

On network matroids and linear network codes

Network coding is a new paradigm in data transport that combines coding with data propagation over a network. Theory of linear network coding (LNC) adopts a linear coding scheme at every node of the network and promises the optimal data transmission rate from the source to all receivers. Linearity enhances the theoretic elegance and engineering simplicity, which leads to wide applicability. This paper reviews the basic theory of LNC and construction algorithms for optimal linear network codes. Exemplifying applications are presented, including random LNC. The fundamental theorem of LNC applies to only acyclic networks, but practical applications actually ignore the acyclic restriction. The theoretic justification for this involves convolutional network coding (CNC), which, however, incurs the difficulty of precise synchronization. The problem can be alleviated when CNC is generalized by selecting an appropriate structure in commutative algebra for data units. This paper tries to present the necessary algebraic concepts as much as possible in engineering language.

Revenue Maximization for Communication Networks with Usage-Based Pricing

Convolutional network coding deals with the propagation of a message pipeline through a cyclic network. We formulate a Convolutional network code by associating every pair of adjacent channels with a rational power series over the base field, called the local encoding kernel, and every channel with a concomitant global encoding kernel, which is a vector of rational power series. Given a complete set of local encoding kernels, a close-form formula is derived for calculating the global encoding kernels. A convolutional multicast is a convolutional network code that every qualified receiving node can decode the message. We offer a construction algorithm for a convolutional multicast as well as a decoding algorithm