Int. J. Electron. Commun. (AEÜ) 68 (2014) 611–615 Contents lists available at ScienceDirect International Journal of Electronics and Communications (AEÜ) j ourna l h om epage: www.elsevier.com/locate/aeue Intra-layer network coding for lossy communication of progressive codes Nima Sarshar, Abdul Bais ∗ Faculty of Engineering, University of Regina, 3737 Wascana Parkway, Regina, SK, Canada a r t i c l e i n f o Article history: Received 15 April 2013 Accepted 10 January 2014 Keywords: Multiple description coding Progressive coding Network coding Layered multicast Linear layered multicast a b s t r a c t In this paper we discuss layered multicast (LM) of progressive source codes using network coding. LM is absolutely optimal if different sinks in the network are satisfied up to their max-flow. Since absolutely optimal intra-layer network strategies might not exist for general networks, we present conditions under which an absolutely optimal, intra-layer multicast strategy exists for a given network and how that strategy may be efficiently constructed. We also discuss the problem of designing optimal intra-layer multicast strategies for general directed networks. © 2014 Elsevier GmbH. All rights reserved. 1. Introduction Multiple description codes can effectively utilize network resources and is able to combat packet loss in communication networks [1]. The network communication throughput can be improved with network coding as compared to routing only strate- gies [2,3]. The problem of lossy source communication using network coding becomes very complex in its most general form [3–5]. A practical subclass of the problem uses progressive source codes along with carefully optimized network coding strategies for efficient multicast of compressible sources [6,5], and is the subject of this paper. We start by introducing a generalization of network coding problem, called Rainbow Network Coding (RNC) [7]. RNC recog- nizes the fact that the information communicated to different members of a multicast group can be different in general. Take the scenario depicted in Fig. 1 as an example. Here, two information bits are to be communicated from node 1 (source) to the sink nodes. The capacity of all links is one. First, lets assume that nodes 4, 5 are sinks. By network coding theorem [2], two bits a, b can be commu- nicated to nodes 5, 6 simultaneously as indicated in Fig. 1(a). Here, transcoding at relay nodes plays a crucial role. In particular, node 7 combines the two bits it receives to produce a ⊕ b (where ⊕ indi- cates XOR operation) which enables nodes 4, 5 to recover both bits. It is easy to verify that at most 3 bits can be communicated to the nodes 4, 5 using routing only. Therefore, the multicast throughput ∗ Corresponding author. Tel.: +1 306 337 8522; fax: +1 306 585 4855. E-mail addresses: nima.sarshar@uregina.ca (N. Sarshar), abdul.bais@uregina.ca (A. Bais). with routing only is at most 1.5 bits per second per node, when the set of sink nodes is T = {4, 5} as shown in Fig. 1(b). Now, what if the set of sink nodes is T = {4, 5, 6}? Since the min- cut (and hence max-flow) into node 6 is only one, the multicast capacity of the network with sinks {4, 5, 6} is only one. Therefore, only one bit of common information per second can be communi- cated to each sink. Both examples in Fig. 1, however, are able to communicate a total of 4 information bits to nodes 4, 5, 6. In Fig. 1(a), nodes 4, 5 each receive 2 bits, while node 6 does not receive any (a total of 4 bits). In contrast, in Fig. 1(b), nodes 4, 6 each receive one bit, while node 5 receives two bits (again, a total of 4 bits). Note that in this case, node 4 is not satisfied up to its max-flow. We can clearly see from this example that all the three nodes in T cannot be satisfied up to their max-flow and hence an absolutely optimal LM strategy cannot be constructed. If each atomic data entity for transmission is assigned a unique “color”, the above problem is about delivering to each node with as large a “spectrum” of colors as possible, without requiring all sink nodes to have exactly the same set of colors. Due to this analogy, we call this form of network information flow problem RNC, a general- ization of rainbow network flow (RNF) to the case where network coding is allowed [8]. Both examples in Fig. 1 are valid rainbow network codes. In Fig. 1(a), the RNC involves transcoding of information at the relay node 7 (the XOR operation). The example in Fig. 1(b), however, uses routing only. This subclass of RNC is called RNF [8]. In this paper, we discuss the idea of layered multicast (LM) [6,5]. In LM, progressive or layered source codes are used for source compression, while network coding is used to multicast these coding layers [3]. The receivers that receive more layers are able to reconstruct the source at lower distortions. The multicast 1434-8411/$ – see front matter © 2014 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.aeue.2014.01.004