Network Coding Fundamentals

Network coding is an elegant and novel technique introduced at the turn of the millennium to improve network throughput and performance. It is expected to be a critical technology for networks of the future. This tutorial addresses the first most natural questions one would ask about this new technique: how network coding works and what are its benefits, how network codes are designed and how much it costs to deploy networks implementing such codes, and finally, whether there are methods to deal with cycles and delay that are present in all real networks. A companion issue deals primarily with applications of network coding

Construction algorithm for network error-correcting codes attaining the Singleton bound

We give a centralized deterministic algorithm for constructing linear network error-correcting codes that attain the Singleton bound of network error-correcting codes. The proposed algorithm is based on the algorithm by Jaggi et al. We give estimates on the time complexity and the required symbol size of the proposed algorithm. We also estimate the probability of a random choice of local encoding vectors by all intermediate nodes giving a network error-correcting codes attaining the Singleton bound. We also clarify the relationship between the robust network coding and the network error-correcting codes with known locations of errors.Comment: To appear in IEICE Trans. Fundamentals (http://ietfec.oxfordjournals.org/), vol. E90-A, no. 9, Sept. 2007. LaTeX2e, 7 pages, using ieice.cls and pstricks.sty. Version 4 adds randomized construction of network error-correcting codes, comparisons of the proposed methods to the existing methods, additional explanations in the proo

Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities

This monograph presents a unified treatment of single- and multi-user problems in Shannon's information theory where we depart from the requirement that the error probability decays asymptotically in the blocklength. Instead, the error probabilities for various problems are bounded above by a non-vanishing constant and the spotlight is shone on achievable coding rates as functions of the growing blocklengths. This represents the study of asymptotic estimates with non-vanishing error probabilities. In Part I, after reviewing the fundamentals of information theory, we discuss Strassen's seminal result for binary hypothesis testing where the type-I error probability is non-vanishing and the rate of decay of the type-II error probability with growing number of independent observations is characterized. In Part II, we use this basic hypothesis testing result to develop second- and sometimes, even third-order asymptotic expansions for point-to-point communication. Finally in Part III, we consider network information theory problems for which the second-order asymptotics are known. These problems include some classes of channels with random state, the multiple-encoder distributed lossless source coding (Slepian-Wolf) problem and special cases of the Gaussian interference and multiple-access channels. Finally, we discuss avenues for further research.Comment: Further comments welcom

Channel coding for network communication: an information theoretic perspective

2011 Fall.Includes bibliographical references.Channel coding helps a communication system to combat noise and interference by adding "redundancy" to the source message. Theoretical fundamentals of channel coding in point-to-point systems have been intensively studied in the research area of information theory, which was proposed by Claude Shannon in his celebrated work in 1948. A set of landmark results have been developed to characterize the performance limitations in terms of the rate and the reliability tradeoff bounds. However, unlike its success in point-to-point systems, information theory has not yielded as rich results in network communication, which has been a key research focus over the past two decades. Due to the limitations posed by some of the key assumptions in classical information theory, network information theory is far from being mature and complete. For example, the classical information theoretic model assumes that communication parameters such as the information rate should be jointly determined by all transmitters and receivers. Communication should be carried out continuously over a long time such that the overhead of communication coordination becomes negligible. The communication channel should be stationary in order for the coding scheme to transform the channel noise randomness into deterministic statistics. These assumptions are valid in a point-to-point system, but they do not permit an extensive application of channel coding in network systems because they have essentially ignored the dynamic nature of network communication. Network systems deal with bursty message transmissions between highly dynamic users. For various reasons, joint determination of key communication parameters before message transmission is often infeasible or expensive. Communication channels can often be non-stationary due to the dynamic communication interference generated by the network users. The objective of this work is to extend information theory toward network communication scenarios. We develop new channel coding results, in terms of the communication rate and error performance tradeoff, for several non-classical communication models, in which key assumptions made in classical channel coding are dropped or revised

Relative Generalized Rank Weight of Linear Codes and Its Applications to Network Coding

By extending the notion of minimum rank distance, this paper introduces two new relative code parameters of a linear code C_1 of length n over a field extension and its subcode C_2. One is called the relative dimension/intersection profile (RDIP), and the other is called the relative generalized rank weight (RGRW). We clarify their basic properties and the relation between the RGRW and the minimum rank distance. As applications of the RDIP and the RGRW, the security performance and the error correction capability of secure network coding, guaranteed independently of the underlying network code, are analyzed and clarified. We propose a construction of secure network coding scheme, and analyze its security performance and error correction capability as an example of applications of the RDIP and the RGRW. Silva and Kschischang showed the existence of a secure network coding in which no part of the secret message is revealed to the adversary even if any dim C_1-1 links are wiretapped, which is guaranteed over any underlying network code. However, the explicit construction of such a scheme remained an open problem. Our new construction is just one instance of secure network coding that solves this open problem.Comment: IEEEtran.cls, 25 pages, no figure, accepted for publication in IEEE Transactions on Information Theor