Index coding studies multiterminal source-coding problems where a set of receivers are required to decode multiple (possibly different) messages from a common broadcast, and they each know some messages a priori. In this paper, at the receiver end, we consider a special setting where each receiver knows only one message a priori, and each message is known to only one receiver. At the broadcasting end, we consider a generalized setting where there could be multiple senders, and each sender knows a subset of the messages. The senders collaborate to transmit an index code. This paper looks at minimizing the number of total coded bits the senders are required to transmit. When there is only one sender, we propose a pruning algorithm to find a lower bound on the optimal (i.e., the shortest) index codelength, and show that it is achievable by linear index codes. When there are two or more senders, we propose an appending technique to be used in conjunction with the pruning technique to give a lower bound on the optimal index codelength; we also derive an upper bound based on cyclic codes. While the two bounds do not match in general, for the special case where no two distinct senders know any message in common, the bounds match, giving the optimal index codelength. The results are expressed in terms of strongly connected components in directed graphs that represent the index-coding problems.

The multiway relay channel is a multicast network where L users exchange data through a relay. In this paper, the capacity region of a class of multiway relay channels is derived, where the channel inputs and outputs take values over finite fields. The cut-set upper bound to the capacity region is derived and is shown to be achievable by our proposed functional-decode-forward coding strategy. More specifically, for the general case where the users can transmit at possibly different rates, functional-decode-forward, combined with rate splitting and joint source-channel decoding, is proved to achieve the capacity region; while for the case where all users transmit at a common rate, rate splitting and joint source-channel decoding are not required to achieve the capacity. That the capacity-achieving coding strategies do not utilize the users' received signals in the users' encoding functions implies that feedback does not increase the capacity region of this class of multiway relay channels.

This paper studies a special class of multicast index coding problems where a sender transmits messages to multiple receivers, each with some side information. Here, each receiver knows a unique message a priori, and there is no restriction on how many messages each receiver requests from the sender. For this class of multicast index coding problems, we obtain the optimal index code, which has the shortest codelength for which the sender needs to send in order for all receivers to obtain their (respective) requested messages. This is the first class of index coding problems where the optimal index codes are found. In addition, linear index codes are shown to be optimal for this class of index coding problems.

We consider a graphical approach to index coding. As cycles have been shown to provide coding gain, cycles and cliques (a specific type of overlapping cycles) have been exploited in an existing literature. In this paper, we define a more general form of overlapping cycles, called the interlinked-cycle (IC) structure, that generalizes cycles and cliques. We propose a scheme, called the interlinked-cycle-cover (ICC) scheme, that leverages IC structures in digraphs to construct scalar linear index codes. We characterize a class of infinitely many digraphs where our proposed scheme is optimal over all linear and nonlinear index codes. Consequently, for this class of digraphs, we indirectly prove that scalar linear index codes are optimal. Furthermore, we show that the ICC scheme can outperform all the existing graph-based schemes (including partial-clique-cover and fractional-local-chromatic number schemes), and a random coding scheme (namely, composite coding) for certain graphs.

The $L$-user additive white Gaussian noise multiway relay channel is investigated, where $L$ users exchange information at the same rate through a single relay. A new achievable rate region, based on the functional-decode-forward coding strategy, is derived. For the case where there are three or more users, and all nodes transmit at the same power, the capacity is obtained. For the case where the relay power scales with the number of users, it is shown that both compress-forward and functional-decode-forward achieve rates within a constant number of bits of the capacity at all SNR levels; in addition, functional-decode-forward outperforms compress-forward and complete-decode-forward at high SNR levels.

The L-user additive white Gaussian noise multi-way relay channel is considered, where multiple users exchange information through a single relay at a common rate. Existing coding strategies, i.e., complete-decode-forward and compress-forward are shown to be bounded away from the cut-set upper bound at high signal-to-noise ratios (SNR). It is known that the gap between the compress-forward rate and the capacity upper bound is a constant at high SNR, and that between the complete-decode-forward rate and the upper bound increases with SNR at high SNR. In this paper, a functional-decode-forward coding strategy is proposed. It is shown that for L ≥ 3, complete-decode-forward achieves the capacity when SNR ≤ 0 dB, and functional-decode-forward achieves the capacity when SNR ≥ 0 dB. For L = 2, functional-decode-forward achieves the capacity asymptotically as SNR increases.