A sender holds a word x consisting of n blocks x i , each of t bits, and wishes to broadcast a codeword to m receivers, R 1 , ..., R m . Each receiver R i is interested in one block, and has prior side information consisting of some subset of the other blocks. Let β t be the minimum number of bits that has to be transmitted when each block is of length t, and let β be the limit β = lim t→∞ β t /t. Informally, β is the average communication cost per bit in each block (for long blocks). Finding the coding rate β, for such an informed broadcast setting, generalizes several coding theoretic parameters related to Informed Source Coding on Demand, Index Coding and Network Coding. In this work we show that usage of large data blocks may strictly improve upon the trivial encoding which treats each bit in the block independently. To this end, we provide general bounds on β t , and prove that for any constant C there is an explicit broadcast setting in which β = 2 but β 1 > C. One of these examples answers a question of [15] . In addition, we provide examples with the following counterintuitive direct-sum phenomena. Consider a union of several mutually independent broadcast settings. The optimal code for the combined setting may yield a significant saving in communication over concatenating optimal encodings for the individual settings. This result also provides new non-linear coding schemes which improve upon the largest known gap between linear and non-linear Network Coding, thus improving the results of [8] . The proofs are based on a relation between this problem and results in the study of Witsenhausen's rate, OR graph products, colorings of Cayley graphs, and the chromatic numbers of Kneser graphs.
A slightly different model is considered in REF , where a sender wishes to broadcast blocks of data to multiple receivers, with each receiver having prior side-information consisting of some subset of the other blocks.
Broadcasting with Side Information
2008 49th Annual IEEE Symposium on Foundations of Computer Science
2008 49th Annual IEEE Symposium on Foundations of Computer Science
Mathematics Computer Science