Conference Paper

# Index Coding with Side Information

Dept. of Electr. Eng., Technion-Israel Inst. of Technol., Haifa

DOI: 10.1109/FOCS.2006.42 Conference: Foundations of Computer Science, 2006. FOCS '06. 47th Annual IEEE Symposium on Source: IEEE Xplore

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**ABSTRACT:**The transmission of packets is considered from one source to multiple receivers over single-hop erasure channels. The objective is to evaluate the stability properties of different transmission schemes with and without network coding. First, the throughput limitation of retransmission schemes is discussed and the stability benefits are shown for randomly coded transmissions, which, however, need not optimize the stable throughput for finite coding field size and finite packet block size. Next, a dynamic scheme is introduced for distributing packets among virtual queues depending on the channel feedback and performing linear network coding based on the instantaneous queue contents. The difference of the maximum stable throughput from the min-cut rate is bounded as function of the order of erasure probabilities depending on the complexity allowed for network coding and queue management. This queue-based network coding scheme can asymptotically optimize the stable throughput to the max-flow min-cut bound, as the erasure probabilities go to zero. This is realized for a finite coding field size without accumulating packet blocks at the source to start network coding. The comparison of random and queue-based dynamic network coding with plain retransmissions opens up new questions regarding the tradeoffs of stable throughput, packet delay, overhead, and complexity.IEEE Transactions on Information Theory 01/2010; · 2.62 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**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={x <sub>1</sub> ,...,x <sub>k</sub> } 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.IEEE Transactions on Information Theory 08/2010; · 2.62 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The following source coding problem was introduced by Birk and Kol: a sender holds a word x isin {0, 1}<sup>n</sup>, and wishes to broadcast a codeword to n receivers, R<sub>n</sub>,..., R<sub>n</sub>. The receiver R<sub>i</sub> is interested in x<sub>i</sub>, and has prior side information comprising some subset of the n bits. This corresponds to a directed graph G on n vertices, where i j is an edge R<sub>i</sub> Ri knows the bit x<sub>j</sub>. An index code for G is an encoding scheme which enables each Ri to always reconstruct x<sub>i</sub>, given his side information. The minimal word length of an index code was studied by Bar-Yossef, Birk, Jayram, and Kol (FOCS'06). They introduced a graph parameter, minrk<sub>2</sub>(G), which completely characterizes the length of an optimal linear index code for G. They showed that in various cases linear codes attain the optimal word length, and conjectured that linear index coding is in fact always optimal. In this work, we disprove the main conjecture of Bar-Yossef, Birk, Jayram, and Kol in the following strong sense: for any epsiv > 0 and sufficiently large n, there is an n-vertex graph G so that every linear index code for G requires codewords of length at least n<sup>epsiv</sup> and yet a nonlinear index code for G has a word length of ne. This is achieved by an explicit construction, which extends Alon's variant of the celebrated Ramsey construction of Frankl and Wilson. In addition, we study optimal index codes in various, less restricted, natural models, and prove several related properties of the graph parameter minrk(G).IEEE Transactions on Information Theory 09/2009; · 2.62 Impact Factor

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