Linear Network Coding For Wireline And Wireless Networks
Abstract
Network Coding is a technique which looks beyond traditional storeandforward 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 [1], where they also demonstrated its use in case of singlesource multiplesink network multicast, although with use of much complex mathematical apparatus. In [1], 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 storeandforward approach. The main result in [1], i.e. the capacity of singlesource multiplesinks information network is nothing but the minimum of the maxﬂows from source to each sink, was again proved by Li, Yeung, and Cai in [2] where they showed that only linear operations suﬃce to achieve the capacity of multicast network. The authors in [2] 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 [3] developed an algebraic framework using tools from algebraic geometry which also proved the multicast maxﬂow mincut theorem proved in [1] and [2]. 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 [3] also dealt with the general linear network coding problem involving multiple sources and multiple sinks with nonuniform 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 [5] 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 [6], 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 [3] or [6]. It was a simultaneous discovery by us(as part of this thesis) as well as by Dougherty, Freiling, Zeger in [7] to determine the characteristics algorithmically.
Applications of Network Coding techniques to wireless networks are seen in literature( [8], [9], [10]), where [8] provided a variant of maxﬂow mincut theorem for wireless networks in the form of linear programming constraints. A new architecture termed as COPE was introduced in [10] which used opportunistic listening and opportunistic coding in wireless mesh networks.
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