Search

For a (single-source) multicast network, the size of a base field is the most known and studied algebraic identity that is involved in characterizing its linear solvability over the base field. In this paper, we design a new class \mathcal N of multicast networks and obtain an explicit formula for the linear solvability of these networks, which involves the associated coset numbers of a multiplicative subgroup in a base field. The concise formula turns out to be the first that matches the topological structure of a multicast network and algebraic identities of a field other than size. It further facilitates us to unveil infinitely many new multicast networks linearly solvable over GF( q ) but not over GF( q' ) with q < q' , based on a subgroup order criterion. In particular: 1) for every k\geq 2 ) but not over GF( 2^{2k+1} ) and 2) for arbitrary distinct primes p and p' , there are infinitely many k and k' ) but not over GF( p'^k' ) with p^k < p'^k' . [ABSTRACT FROM PUBLISHER]

The fundamental result of linear network coding asserts the existence of an optimal code on an acyclic single-source multicast network when the symbol field is sufficiently large. The restriction to acyclic networks turns out to stem from the customary structure of the symbol alphabet as a field. Adopting data units belonging to a discrete valuation ring (DVR), that is, a PID with a unique maximal ideal, much of the network coding theory extends to cyclic single-source multicast networks. Convolutional network coding is the instance of DVR-based network coding when the DVR consists of rational power series over the symbol field. Meanwhile, a field can be regarded as a degenerate DVR since it is a PID with the maximal ideal 0. Thus the conventional field-based network coding theory becomes a degenerate version of the DVR-based theory. This paper also delves into the issue of constructing optimal network codes on cyclic networks. Inspired by matroid duality theory, a novel method is devised to take advantage of all existing acyclic algorithms for network code construction. It associates every cyclic network with a quadratically large acyclic network so that essentially every optimal code on the acyclic network directly induces one on the cyclic network. [ABSTRACT FROM AUTHOR]

In order to guarantee data reliability, erasure codes have been used in distributed storage systems. Nevertheless, this mechanism suffers from the repair problem that excess data are needed to repair a single failure, causing both high bandwidth consuming for the network and heavy computing load on the replacement node. To reduce repair traffic, researchers pointed out the tradeoff between storage and repair traffic and proposed regenerating codes by combining network coding. However, the combination only focuses on the storage terminal and the construction of the codes is quite complicated. Therefore, this paper further combines network coding with network structure and proposes a repair tree model based on general erasure codes to simplify the repair procedure. By decomposing repair computing and distributing it among the tree nodes, our model can mitigate the computing tension. The performance of repair tree is analyzed and evaluated by preliminary emulation. The result shows it can make about three times faster computing than conventional measure and the repair throughput is doubled if there are network bottlenecks. For proper topology, it can significantly reduce the repair traffic. We present algorithms to generate trees across the network topology. At last, we present the idea of extending repair tree to repair multiple failures. [ABSTRACT FROM AUTHOR]