Linear Network Coding For Wireline And Wireless Networks
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Network Coding is a technique which looks beyond traditional store-and-forward approach followed by routers and switches in communication networks, and as an extension introduces maps termed as ‘local encoding kernels’ and ‘global encoding kernels’ deﬁned for each communication link in the network. The purpose of both these maps is to deﬁne rules as to how to combine the packets input on the node to form a packet going out on an edge. The paradigm of network coding was formally and for the ﬁrst time introduced by Ahlswede et al. in , where they also demonstrated its use in case of single-source multiple-sink network multicast, although with use of much complex mathematical apparatus. In , examples of networks are also presented where it is shown that network coding can improve the overall throughput of the network which can not otherwise be realized by the conventional store-and-forward approach. The main result in , i.e. the capacity of single-source multiple-sinks information network is nothing but the minimum of the max-ﬂows from source to each sink, was again proved by Li, Yeung, and Cai in  where they showed that only linear operations suﬃce to achieve the capacity of multicast network. The authors in  deﬁned generalizations to the multicast problem, which they termed as linear broadcast, linear dispersion, and Generic LCM as strict generalizations of linear multicast, and showed how to build linear network codes for each of these cases. For the case of linear multicast, Koetter and Medard in  developed an algebraic framework using tools from algebraic geometry which also proved the multicast max-ﬂow min-cut theorem proved in  and . It was shown that if the size of the ﬁnite ﬁeld is bigger than a certain threshold, then there always exists a solution to the linear multicast, provided it is solvable. In other words, a solvable linear multicast always has a solution in any ﬁnite ﬁeld whose cardinality is greater than the threshold value. The framework in  also dealt with the general linear network coding problem involving multiple sources and multiple sinks with non-uniform demand functions at the sinks, but did not touched upon the key problem of ﬁnding the characteristic(s) of the ﬁeld in which it may have solution. It was noted in  that a solvable network may not have a linear solution at all, and then introduced the notion of general linear network coding, where the authors conjectured that every solvable network must have a general linear solution. This was refuted by Dougherty, Freiling, Zeger in , where the authors explicitly constructed example of a solvable network which has no general linear solution, and also networks which have solution in a ﬁnite ﬁeld of char 2, and not in any other ﬁnite ﬁeld. But an algorithm to ﬁnd the characteristic of the ﬁeld in which a scalar or general linear solution(if at all) exists did not ﬁnd any mention in  or . It was a simultaneous discovery by us(as part of this thesis) as well as by Dougherty, Freiling, Zeger in  to determine the characteristics algorithmically. Applications of Network Coding techniques to wireless networks are seen in literature( , , ), where  provided a variant of max-ﬂow min-cut theorem for wireless networks in the form of linear programming constraints. A new architecture termed as COPE was introduced in  which used opportunistic listening and opportunistic coding in wireless mesh networks.
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