In the index coding problem a sender holds a message @math and wishes to broadcast information to @math receivers in a way that enables the @math th receiver to retrieve the @math th bit @math . Every receiver has prior side information comprising a subset of the bits of @math , and the goal is to minimize the length of the information sent via the broadcast channel. Porter and Wootters have recently introduced the model of embedded index coding, where the receivers also play the role of the sender and the goal is to minimize the total length of their broadcast information. An embedded index code is said to be task-based if every receiver retrieves its bit based only on the information provided by one of the receivers. This short paper studies the effect of the task-based restriction on linear embedded index coding. It is shown that for certain side information maps there exists a linear embedded index code of length quadratically smaller than that of any task-based embedded index code. The result attains, up to a multiplicative constant, the largest possible gap between the two quantities. The proof is by an explicit construction and the analysis involves spectral techniques.
The index coding problem, introduced in @cite_0 and further developed in @cite_7 , has been studied in various variations and extensions. This research is motivated by applications such as distributed storage @cite_2 , wireless communication @cite_16 , and the more general problem of network coding @cite_8 . The variant called embedded index coding, introduced in @cite_17 , can be viewed as a special case of the multi-sender index coding model studied in @cite_4 which allows multiple senders and multiple receivers but as disjoint sets of vertices (see also @cite_14 ). The framework of index coding studied in @cite_17 is more general than the one considered in the current work and allows the receivers to request multiple messages.
@cite_14 @cite_4 @cite_7 @cite_8 @cite_0 @cite_2 @cite_16 @cite_17
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The index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless ad hoc networks. An instance of the index coding problem includes a sender that holds a set of information messages X= x1,...,xk and a set of receivers R. Each receiver (x,H) in R needs to obtain a message x X and has prior side information consisting of a subset H of X . The sender uses a noiseless communication channel to broadcast encoding of messages in X to all clients. The objective is to find an encoding scheme that minimizes the number of transmissions required to satisfy the demands of all the receivers. In this paper, we analyze the relation between the index coding problem, the more general network coding problem, and the problem of finding a linear representation of a matroid. In particular, we show that any instance of the network coding and matroid representation problems can be efficiently reduced to an instance of the index coding problem. Our reduction implies that many important properties of the network coding and matroid representation problems carry over to the index coding problem. Specifically, we show that vector linear codes outperform scalar linear index codes and that vector linear codes are insufficient for achieving the optimum number of transmissions. 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. Author(s): Maleki, H; Cadambe, VR; Jafar, SA | Abstract: The index coding problem is studied from an interference alignment perspective providing new results as well as new insights into, and generalizations of, previously known results. An equivalence is established between the capacity of multiple unicast index coding (where each message is desired by exactly one receiver), and groupcast index coding (where a message can be desired by multiple receivers), which settles the heretofore open question of insufficiency of linear codes for the multiple unicast index coding problem by equivalence with groupcast settings, where this question has previously been answered. Necessary and sufficient conditions for the achievability of rate half per message in the index coding problem are shown to be a natural consequence of interference alignment constraints, and generalizations to feasibility of rate 1 (L+1)per message when each destination desires at least L messages, are similarly obtained. Finally, capacity optimal solutions are presented to a series of symmetric index coding problems inspired by the local connectivity and local interference characteristics of wireless networks. The solutions are based on vector linear coding. © 1963-2012 IEEE. We consider a Multi-Sender Unicast Index-Coding (MSUIC) problem, where in a broadcast network, multiple senders collaboratively send distinct messages to multiple receivers, each having some subset of the messages a priori . The aim is to find the shortest index code that minimizes the total number of coded bits sent by the senders. In this paper, built on the classic single-sender minrank concept, we develop a new rank-minimization framework for MSUIC that explicitly takes into account the sender message constraints and minimizes the sum of the ranks of encoding matrices subject to the receiver decoding requirements. This framework provides a systematic way to construct multi-sender linear index codes and to study their capability in achieving the shortest index codelength per message length (i.e., the optimal broadcast rate). In particular, we establish the optimal broadcast rate for all critical MSUIC instances with up to four receivers and show that a binary linear index code is optimal for all, except 15 instances with four receivers. We also propose a heuristic algorithm (in lieu of exhaustive search) to solve the rank-minimization problem. The effectiveness of the algorithm is validated by numerical studies of MSUIC instances with four or more receivers.