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17
A rankmetric approach to error control in random network coding
 IEEE Transactions on Information Theory
"... It is shown that the error control problem in random network coding can be reformulated as a generalized decoding problem for rankmetric codes. This result allows many of the tools developed for rankmetric codes to be applied to random network coding. In the generalized decoding problem induced by ..."
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Cited by 159 (11 self)
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It is shown that the error control problem in random network coding can be reformulated as a generalized decoding problem for rankmetric codes. This result allows many of the tools developed for rankmetric codes to be applied to random network coding. In the generalized decoding problem induced by random network coding, the channel may supply partial information about the error in the form of erasures (knowledge of an error location but not its value) and deviations (knowledge of an error value but not its location). For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can fully exploit the correction capability of the code; namely, it can correct any pattern of ǫ errors, µ erasures and δ deviations provided 2ǫ+ µ + δ ≤ d − 1, where d is the minimum rank distance of the code. Our approach is based on the coding theory for subspaces introduced by Koetter and Kschischang and can be seen as a practical way to construct codes in that context. I.
On metrics for error correction in network coding
 IEEE Trans. Inf. Theory
, 2009
"... The problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model. For coherent network coding, where knowledge of the network topology and network code is assumed at the source and destination nodes, the error correction capability of an (outer ..."
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Cited by 41 (4 self)
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The problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model. For coherent network coding, where knowledge of the network topology and network code is assumed at the source and destination nodes, the error correction capability of an (outer) code is succinctly described by the rank metric; as a consequence, it is shown that universal network error correcting codes achieving the Singleton bound can be easily constructed and efficiently decoded. For noncoherent network coding, where knowledge of the network topology and network code is not assumed, the error correction capability of a (subspace) code is given exactly by a new metric, called the injection metric, which is closely related to, but different than, the subspace metric of Kötter and Kschischang. In particular, in the case of a nonconstantdimension code, the decoder associated with the injection metric is shown to correct more errors then a minimumsubspacedistance decoder. All of these results are based on a general approach to adversarial error correction, which could be useful for other adversarial channels beyond network coding. Index Terms Adversarial channels, error correction, injection distance, network coding, rank distance, subspace codes.
On wiretap networks II
 IN PROC. IEEE INT. SYMP. INFORMATION THEORY
, 2008
"... We consider the problem of securing a multicast network against a wiretapper that can intercept the packets on a limited number of arbitrary network links of his choice. We assume that the network implements network coding techniques to simultaneously deliver all the packets available at the source ..."
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Cited by 27 (5 self)
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We consider the problem of securing a multicast network against a wiretapper that can intercept the packets on a limited number of arbitrary network links of his choice. We assume that the network implements network coding techniques to simultaneously deliver all the packets available at the source to all the destinations. We show how this problem can be looked at as a network generalization of the OzarowWyner Wiretap Channel of type II. In particular, we show that network security can be achieved by using the OzarowWyner approach of coset coding at the source on top of the implemented network code. This way, we quickly and transparently recover some of the results available in the literature on secure network coding for wiretapped networks. We also derive new bounds on the required secure code alphabet size and an algorithm for code construction.
Secure Network Coding for Wiretap Networks of Type II
, 2009
"... We consider the problem of securing a multicast network against a wiretapper that can intercept the packets on a limited number of arbitrary network edges of its choice. We assume that the network employs the network coding technique to simultaneously deliver the packets available at the source to a ..."
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Cited by 19 (2 self)
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We consider the problem of securing a multicast network against a wiretapper that can intercept the packets on a limited number of arbitrary network edges of its choice. We assume that the network employs the network coding technique to simultaneously deliver the packets available at the source to all the receivers. We show that this problem can be looked at as a network generalization of the wiretap channel of type II introduced in a seminal paper by Ozarow and Wyner. In particular, we show that the transmitted information can be secured by using the OzarowWyner approach of coset coding at the source on top of the existing network code. This way, we quickly and transparently recover some of the results available in the literature on secure network coding for wiretap networks. Moreover, we derive new bounds on the required alphabet size that are independent of the network size and devise an algorithm for the construction of secure network codes. We also look at the dual problem and analyze the amount of information that can be gained by the wiretapper as a function of the number of wiretapped edges.
Refined coding bounds for network error correction
 in Proc. IEEE Information Theory Workshop 2007
, 2007
"... Abstract—With respect to a given set of local encoding kernels defining a linear network code, refined versions of the Hamming bound, the Singleton bound and the GilbertVarshamov bound for network error correction are proved by the weight properties of network codes. This refined Singleton bound is ..."
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Cited by 16 (2 self)
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Abstract—With respect to a given set of local encoding kernels defining a linear network code, refined versions of the Hamming bound, the Singleton bound and the GilbertVarshamov bound for network error correction are proved by the weight properties of network codes. This refined Singleton bound is also proved to be tight for linear message sets. Index Terms—Network error correction coding, network Hamming weight, Singleton bound, GilbertVarshamov bound. I.
Rate regions for coherent and noncoherent multisource network error correction
 in Proc. of IEEE International Symposium of Information Theory
, 2009
"... AbstractIn this paper we derive capacity regions for network error correction with both known and unknown topologies (coherent and noncoherent network coding) under a multiplesource multicast transmission scenario. For the multiplesource nonmulticast scenario, given any achievable network code ..."
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Cited by 10 (6 self)
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AbstractIn this paper we derive capacity regions for network error correction with both known and unknown topologies (coherent and noncoherent network coding) under a multiplesource multicast transmission scenario. For the multiplesource nonmulticast scenario, given any achievable network code for the errorfree case, we construct a code with a reduced rate region for the case with errors.
Network Coding Applications
, 2007
"... 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 deals with wireless and content distribution networks, considered to be the mo ..."
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Cited by 10 (1 self)
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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 deals with wireless and content distribution networks, considered to be the most likely applications of network coding, and it also reviews emerging applications of network coding such as network monitoring and management. Multiple unicasts, security, networks with unreliable links, and quantum networks are also addressed. The preceding companion deals with theoretical foundations of network coding.
Refined Coding Bounds and Code Constructions for Coherent Network Error Correction
, 2009
"... Coherent network error correction is the errorcontrol problem in network coding with the knowledge of the network codes at the source and sink nodes. With respect to a given set of local encoding kernels defining a linear network code, we obtain refined versions of the Hamming bound, the Singleton ..."
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Cited by 9 (0 self)
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Coherent network error correction is the errorcontrol problem in network coding with the knowledge of the network codes at the source and sink nodes. With respect to a given set of local encoding kernels defining a linear network code, we obtain refined versions of the Hamming bound, the Singleton bound and the GilbertVarshamov bound for coherent network error correction. Similar to its classical counterpart, this refined Singleton bound is tight for linear network codes. The tightness of this refined bound is shown by two construction algorithms of linear network codes achieving this bound. These two algorithms illustrate different design methods: one makes use of existing network coding algorithms for errorfree transmission and the other makes use of classical errorcorrecting codes. The implication of the tightness of the refined Singleton bound is that sink nodes with higher maximum flow values can have higher error correction capabilities.
Multipleaccess Network Informationflow and Correction Codes
"... This work considers the multipleaccess multicast errorcorrection scenario over a packetized network with z malicious edge adversaries. The network has mincut m and packets of length ℓ, and each sink demands all information from the set of sources S. The capacity region is characterized for both ..."
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Cited by 4 (3 self)
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This work considers the multipleaccess multicast errorcorrection scenario over a packetized network with z malicious edge adversaries. The network has mincut m and packets of length ℓ, and each sink demands all information from the set of sources S. The capacity region is characterized for both a “sidechannel ” model (where sources and sinks share some random bits that are secret from the adversary) and an “omniscient” adversarial model (where no limitations on the adversary’s knowledge are assumed). In the “sidechannel” adversarial model, the use of a secret channel allows higher rates to be achieved compared to the “omniscient” adversarial model, and a polynomialcomplexity capacityachieving code is provided. For the “omniscient ” adversarial model, two capacityachieving constructions are given: the first is based on random subspace code design and has complexity exponential in ℓm, while
Adversarial error correction for network coding: Models and metrics
 IN PROCEEDINGS OF ANNUAL ALLERTON CONFERENCE ON COMMUNICATIONS, CONTROL, AND COMPUTING
, 2008
"... The problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model. For coherent network coding, where knowledge of the network topology and network code is assumed at the source and destination nodes, the error correction capability of an (oute ..."
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Cited by 2 (0 self)
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The problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model. For coherent network coding, where knowledge of the network topology and network code is assumed at the source and destination nodes, the error correction capability of an (outer) code is succinctly described by the rank metric; as a consequence, it is shown that universal network error correcting codes achieving the Singleton bound can be easily constructed and efficiently decoded. For noncoherent network coding, where knowledge of the network topology and network code is not assumed, the error correction capability of a (subspace) code is given exactly by a modified subspace metric, which is closely related to, but different than, the subspace metric of Kötter and Kschischang. In particular, in the case of a nonconstantdimension code, the decoder associated with the modified metric is shown to correct more errors then a minimum subspace distance decoder.