### 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 |

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

### Other

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

City | Philadelphia, PA |

Period | 10/25/08 → 10/28/08 |

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### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008*(pp. 823-832). [4691014] https://doi.org/10.1109/FOCS.2008.41

**Broadcasting with side information.** / Alon, Noga; Hassidim, Avinatan; Lubetzky, Eyal; Stav, Uri; Weinstein, Amit.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008.*, 4691014, pp. 823-832, 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, Philadelphia, PA, United States, 10/25/08. https://doi.org/10.1109/FOCS.2008.41

}

TY - GEN

T1 - Broadcasting with side information

AU - Alon, Noga

AU - Hassidim, Avinatan

AU - Lubetzky, Eyal

AU - Stav, Uri

AU - Weinstein, Amit

PY - 2008

Y1 - 2008

N2 - A sender holds a word x consisting of n blocks xi, each of t bits, and wishes to broadcast a codeword to m receivers, R1,..., Rm. Each receiver Ri is interested in one block, and has prior side information consisting of some subset of the other blocks. Let βt 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; βt/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 βt, and prove that for any constant C there is an explicit broadcast setting in which β= 2 but β1 > 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.

AB - A sender holds a word x consisting of n blocks xi, each of t bits, and wishes to broadcast a codeword to m receivers, R1,..., Rm. Each receiver Ri is interested in one block, and has prior side information consisting of some subset of the other blocks. Let βt 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; βt/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 βt, and prove that for any constant C there is an explicit broadcast setting in which β= 2 but β1 > 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.

UR - http://www.scopus.com/inward/record.url?scp=57949107168&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57949107168&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2008.41

DO - 10.1109/FOCS.2008.41

M3 - Conference contribution

AN - SCOPUS:57949107168

SN - 9780769534367

SP - 823

EP - 832

BT - Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

ER -