## Abstract

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 _{→Infin;} 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.

Original language | English (US) |
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Title of host publication | Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 |

Pages | 823-832 |

Number of pages | 10 |

DOIs | |

State | Published - 2008 |

Externally published | Yes |

Event | 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 - Philadelphia, PA, United States Duration: Oct 25 2008 → Oct 28 2008 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Other

Other | 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 |
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Country/Territory | United States |

City | Philadelphia, PA |

Period | 10/25/08 → 10/28/08 |

## All Science Journal Classification (ASJC) codes

- Computer Science(all)