Abstract-In this paper, we focus on the fundamental problem of finding the optimal encoding for the broadcasted packets that minimizes the overall number of transmissions. We show that this problem is NP-complete over GF (2) and establish several fundamental properties of the optimal solution. We also propose a simple heuristic solution for the problem based on graph coloring and present some empirical results for random settings.

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 Ozarow-Wyner Wiretap Channel of type II. In particular, we show that network security can be achieved by using the Ozarow-Wyner 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.

We investigate the network coding problem in a minimal size of a finite field over which a linear network code certain class of minimal multicast networks. In a multicast coding exists for a certain multicast problem is NP-hard. network, a source S needs to deliver h symbols, or packets, to aIn this work, we focus on the network coding problem set of destinations T over an underlying communication network for minimal multicast networks. A coding network (G, S, T) modeled by a graph G. A coding network is said to be h-minimal i if it can deliver h symbols from S to the destination nodes, while is said to be h-minimal if it can deliver h packets from any proper subnetwork of G can deliver at most h -1 symbols S to T, while any proper subnetwork of G can deliver to the set of destination nodes. This problem is motivated by at most h -1 packets. The problem is motivated by the the requirement to minimize the amount of network resources need of network service providers to minimize the amount allocated for a multicast connections.. Wllocaed how ta multicast networksohav of network resources allocated for individual multicast We show that, surprisingly, minimal multicast networks have concin. Ined miia. utcs ntok nld unique properties that distinguish them from the general case of connections. Indeed, minimal multcast networks include multicast networks. In particular, we show that it is possible to only links that are essential for delivery of h packets to determine whether a 2-minimal network has a routing solution all T terminals, which minimizes the cost of establishing a (i.e., a solution without encoding nodes) in polynomial time, while multicast connection. this problem is NP-hard in general. In addition, we show that if a 2-minimal network is planar, then the minimum size of the required field for linear network codes is at most 3. Also, weContributions. The contribution of our paper can be suminvestigate several structural properties of 2-minimal networks marized as follows. First, we analyze the complexity of and generalize our results for h > 2. deciding whether a given multicast problem admits a pure routing solution (i.e., a solution that does not require networkcoding). We show that this problem can be solved in linear time for 2-minimal coding networks. For the general case of A fundamental problem in the design of communication non-minimal coding networks, this problem was shown to be networks is to deliver information between the source and the NP-hard [12]. We present here another proof of this result destination nodes. Recently, it was shown that the information based on a reduction from the problem of vertex coloring of delivery can be facilitated by employing the novel technique multigraphs. of network coding [5]. The main idea of network codingWe also show that all network coding problems in 2is to allow intermediate nodes in the network to generate minimal networks have a similar structure. Specifically, any new packets by mixing the information received over their such problem can be reduced to ...

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Abstract-Let Aq(n, d) be the maximum order (maximum number of codewords) of a q-ary code of length n and Hamming distance at least d. And let A(n, d, w) that of a binary code of constant weight w. Building on results from algebraic graph theory and Erdős-ko-Rado like theorems in extremal combinatorics, we show how several known bounds on Aq(n, d) and A(n, d, w) can be easily obtained in a single framework. For instance, both the Hamming and Singleton bounds can derived as an application of a property relating the clique number and the independence number of vertex transitive graphs. Using the same techniques, we also derive some new bounds and present some additional applications.