A New Class of Index Coding Instances Where Linear Coding is Optimal

We study index-coding problems (one sender broadcasting messages to multiple receivers) where each message is requested by one receiver, and each receiver may know some messages a priori. This type of index-coding problems can be fully described by directed graphs. The aim is to find the minimum codelength that the sender needs to transmit in order to simultaneously satisfy all receivers' requests. For any directed graph, we show that if a maximum acyclic induced subgraph (MAIS) is obtained by removing two or fewer vertices from the graph, then the minimum codelength (i.e., the solution to the index-coding problem) equals the number of vertices in the MAIS, and linear codes are optimal for this index-coding problem. Our result increases the set of index-coding problems for which linear index codes are proven to be optimal.Comment: accepted and to be presented at the 2014 International Symposium on Network Coding (NetCod

Linear Codes are Optimal for Index-Coding Instances with Five or Fewer Receivers

We study zero-error unicast index-coding instances, where each receiver must perfectly decode its requested message set, and the message sets requested by any two receivers do not overlap. We show that for all these instances with up to five receivers, linear index codes are optimal. Although this class contains 9847 non-isomorphic instances, by using our recent results and by properly categorizing the instances based on their graphical representations, we need to consider only 13 non-trivial instances to solve the entire class. This work complements the result by Arbabjolfaei et al. (ISIT 2013), who derived the capacity region of all unicast index-coding problems with up to five receivers in the diminishing-error setup. They employed random-coding arguments, which require infinitely-long messages. We consider the zero-error setup; our approach uses graph theory and combinatorics, and does not require long messages.Comment: submitted to the 2014 IEEE International Symposium on Information Theory (ISIT

Optimal Routing for the Gaussian Multiple-Relay Channel with Decode-and-Forward

In this paper, we study a routing problem on the Gaussian multiple relay channel, in which nodes employ a decode-and-forward coding strategy. We are interested in routes for the information flow through the relays that achieve the highest DF rate. We first construct an algorithm that provably finds optimal DF routes. As the algorithm runs in factorial time in the worst case, we propose a polynomial time heuristic algorithm that finds an optimal route with high probability. We demonstrate that that the optimal (and near optimal) DF routes are good in practice by simulating a distributed DF coding scheme using low density parity check codes with puncturing and incremental redundancy.Comment: Accepted and to be presented at the 2007 IEEE International Symposium on Information Theory (ISIT 2007), Acropolis Congress and Exhibition Center, Nice, France, June 24-29 200

Myopic Coding in Multiple Relay Channels

In this paper, we investigate achievable rates for data transmission from sources to sinks through multiple relay networks. We consider myopic coding, a constrained communication strategy in which each node has only a local view of the network, meaning that nodes can only transmit to and decode from neighboring nodes. We compare this with omniscient coding, in which every node has a global view of the network and all nodes can cooperate. Using Gaussian channels as examples, we find that when the nodes transmit at low power, the rates achievable with two-hop myopic coding are as large as that under omniscient coding in a five-node multiple relay channel and close to that under omniscient coding in a six-node multiple relay channel. These results suggest that we may do local coding and cooperation without compromising much on the transmission rate. Practically, myopic coding schemes are more robust to topology changes because encoding and decoding at a node are not affected when there are changes at remote nodes. Furthermore, myopic coding mitigates the high computational complexity and large buffer/memory requirements of omniscient coding.Comment: To appear in the proceedings of the 2005 IEEE International Symposium on Information Theory, Adelaide, Australia, September 4-9, 200

The Capacity Region of Restricted Multi-Way Relay Channels with Deterministic Uplinks

This paper considers the multi-way relay channel (MWRC) where multiple users exchange messages via a single relay. The capacity region is derived for a special class of MWRCs where (i) the uplink and the downlink are separated in the sense that there is no direct user-to-user links, (ii) the channel is restricted in the sense that each user's transmitted channel symbols can depend on only its own message, but not on its received channel symbols, and (iii) the uplink is any deterministic function.Comment: Author's final version (to be presented at ISIT 2012

The Capacity Region of the Restricted Two-Way Relay Channel with Any Deterministic Uplink

This paper considers the two-way relay channel (TWRC) where two users communicate via a relay. For the restricted TWRC where the uplink from the users to the relay is any deterministic function and the downlink from the relay to the users is any arbitrary channel, the capacity region is obtained. The TWRC considered is restricted in the sense that each user can only transmit a function of its message.Comment: author's final version (accepted and to appear in IEEE Communications Letters