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

ABSTRACT Motivated by a problem of transmitting data over broadcast channels (BirkandKol, INFOCOM1998), we study the following coding problem: a sender communicates with n receivers Rl,.., Rn. He holds an input x isin {0, 1}n and wishes to broadcast a single message so that each receiver Ri can recover the bit xi. Each Ri has prior side information about x, induced by a directed graph G on n nodes; Ri knows the bits of x in the positions {j | (i, j) is anedge of G}. We call encoding schemes that achieve this goal INDEX codes for {0, 1} n with side information graph G. In this paper we identify a measure on graphs, the minrank, which we conjecture to exactly characterize the minimum length of INDEX codes. We resolve the conjecture for certain natural classes of graphs. For arbitrary graphs, we show that the minrank bound is tight for both linear codes and certain classes of non-linear codes. For the general problem, we obtain a (weaker) lower bound that the length of an INDEX code for any graph G is at least the size of the maximum acyclic induced subgraph of G

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    • "codewords, x (n) k , generated according to an independent normal distribution X k ∼ N (0, α k P ), where α k ≥0 and k α k = 1 to satisfy the transmission power constraint. Multiplexing coding [16] and index coding [17] are employed to construct the subcodebooks. In multiplexing coding, two or more messages are first bijectively mapped to a single message, and then, the codewords are generated for this message. "
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    ABSTRACT: This paper investigates the capacity region of the three-receiver AWGN broadcast channel where the receivers (i) have private-message requests, and (ii) may know some of the messages requested by other receivers as side information. We first classify all 64 possible side information configurations into eight groups, each consisting of eight members.We next construct transmission schemes, and derive new inner and outer bounds for the groups. This establishes the capacity region for 52 out of 64 possible side information configurations. For six groups (i.e., groups 1, 2, 3, 5, 6, and 8 in our terminology), we establish the capacity region for all their members, and show that it tightens both the best known inner and outer bounds. For group 4, our inner and outer bounds tighten the best known inner bound and/or outer bound for all the group members. Moreover, our bounds coincide at certain regions, which can be characterized by two thresholds. For group 7, our inner and outer bounds coincide for four members, thereby establishing the capacity region. For the remaining four members, our bounds tighten both the best known inner and outer bounds.
    IEEE Transactions on Information Theory 07/2015; DOI:10.1109/TIT.2015.2463277 · 2.65 Impact Factor
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    • "The work of [5] established the connection between linear index codes and the minimum rank of the side information graph representing the problem. The sub-optimality of linear index codes was shown in [18]–[20]. "
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    ABSTRACT: Index codes reduce the number of bits broadcast by a wireless transmitter to a number of receivers with different demands and with side information. It is known that the problem of finding optimal linear index codes is NP-hard. We investigate the performance of different heuristics based on rank minimization and matrix completion methods, such as alternating projections and alternating minimization, for constructing linear index codes over the reals. As a summary of our results, the alternating projections method gives the best results in terms of minimizing the number of broadcast bits and convergence rate and leads to up to 13% savings in average communication cost compared to graph coloring algorithms studied in the literature. Moreover, we describe how the proposed methods can be used to construct linear network codes for non-multicast networks. Our computer code is available online.
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    • "The capacity region C of the index coding problem is defined as the closure of the set of achievable rate tuples. Since Birk and Kol [1] introduced the index coding problem in 1998, this simple yet fundamental problem attracted several research communities (see [2]–[5] for a subset of recent contributions). The capacity region has been established for all 9,608 index coding problems of n = 5 messages [6] (which includes all index coding problems up to five messages by taking projections). "
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    ABSTRACT: The capacity region of the index coding problem is characterized through the notion of confusion graph and its fractional chromatic number. Based on this multiletter characterization, several structural properties of the capacity region are established, some of which are already noted by Tahmasbi, Shahrasbi, and Gohari, but proved here with simple and more direct graph-theoretic arguments. In particular, the capacity region of a given index coding problem is shown to be simple functionals of the capacity regions of smaller subproblems when the interaction between the subproblems is none, one-way, or complete.
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