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Optimal Solution for the Index Coding Problem Using Network Coding over GF(2)
"... Abstract—The index coding problem is a fundamental transmission problem which occurs in a wide range of multicast networks. Network coding over a large finite field size has been shown to be a theoretically efficient solution to the index coding problem. However the high computational complexity of ..."
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Abstract—The index coding problem is a fundamental transmission problem which occurs in a wide range of multicast networks. Network coding over a large finite field size has been shown to be a theoretically efficient solution to the index coding problem. However the high computational complexity of packet encoding and decoding over a large finite field size, and its subsequent penalty on encoding and decoding throughput and higher energy cost makes it unsuitable for practical implementation in processor and energy constraint devices like mobile phones and wireless sensors. While network coding over GF(2) can alleviate these concerns, it comes at a tradeoff cost of degrading throughput performance. To address this tradeoff, we propose a throughput optimal triangular network coding scheme over GF(2). We show that such a coding scheme can supply unlimited number of innovative packets and the decoding involves the simple back substitution. Such a coding scheme provides an efficient solution to the index coding problem and its lower computation and energy cost makes it suitable for practical implementation on devices with limited processing and energy capacity. I.
Linear Network Code for Erasure Broadcast Channel with Feedback: Complexity and Algorithms
, 2013
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Linear Network Code for Erasure Broadcast Channel with Feedback: Complexity and Algorithms
"... AbstractThis paper investigates the construction of linear network codes for broadcasting a set of data packets to a number of users. The links from the source to the users are modeled as independent erasure channels. Users are allowed to inform the source node whether a packet is received correct ..."
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AbstractThis paper investigates the construction of linear network codes for broadcasting a set of data packets to a number of users. The links from the source to the users are modeled as independent erasure channels. Users are allowed to inform the source node whether a packet is received correctly via feedback channels. In order to minimize the number of packet transmissions until all users have received all packets successfully, it is necessary that a data packet, if successfully received by a user, can increase the dimension of the vector space spanned by the encoding vectors he or she has received by one. Such an encoding vector is called innovative. To reduce decoding complexity, sparse encoding vectors are preferred, since the sparsity can be exploited when solving systems of linear equations. Generating a sparsest encoding vector with large finite field size, however, is shown to be NPhard. An approximation algorithm is constructed. For binary field, heuristic algorithms are also proposed. Index TermsErasure broadcast channel, network coding, computational complexity.
2 Optimal Solution for the Index Coding Problem Using Network Coding over GF(2)
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On Deterministic Linear Network Coded Broadcast and Its Relation to Matroid Theory
, 2014
"... Deterministic linear network coding (DLNC) is an important family of network coding techniques for wireless packet broadcast. In this paper, we show that DLNC is strongly related to and can be effectively studied using matroid theory without bridging index coding. We prove the equivalence between t ..."
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Deterministic linear network coding (DLNC) is an important family of network coding techniques for wireless packet broadcast. In this paper, we show that DLNC is strongly related to and can be effectively studied using matroid theory without bridging index coding. We prove the equivalence between the DLNC solution and matrix matroid. We use this equivalence to study the performance limits of DLNC in terms of the number of transmissions and its dependence on the finite field size. Specifically, we derive the sufficient and necessary condition for the existence of perfect DLNC solutions and prove that such solutions may not exist over certain finite fields. We then show that identifying perfect solutions over any finite field is still an open problem in general. To fill this gap, we develop a heuristic algorithm which employs graphic matroids to find perfect DLNC solutions over any finite field. Numerical results show that its performance in terms of minimum number of transmissions is close to the lower bound, and is better than random linear network coding when the field size is not so large.