### Abstract

The following source coding problem was introduced by Birk and Kol: a sender holds a word x ∈ {0, 1}^{n}, and wishes to broadcast a codeword to n receivers, R_{1}, . . . , R_{n}. The receiver R_{1} is interested in x_{i}, and has prior side information comprising some subset of the n bits. This corresponds to a directed graph G on n vertices, where ij is an edge iff R_{i} knows the bit x_{j}. An index code for G is an encoding scheme which enables each R_{i} to always reconstruct x_{i}, given his side information. The minimal word length of an index code was studied by Bar-Yossef, Birk, Jayram and Kol [4], They introduced a graph parameter, minrk_{2}(G), which completely characterizes the length of an optimal linear index code for G The authors of [4] 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 [4] in the following strong sense: for any ε > 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 ^{1-ε}, and yet a non-linear index code for G has a word length of n^{ε}. This is achieved by an explicit construction, which extends Alon's variant of the celebrated Ramsey construction of Frankl and Wilson.

Original language | English (US) |
---|---|

Title of host publication | Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007 |

Pages | 161-168 |

Number of pages | 8 |

DOIs | |

State | Published - 2007 |

Event | 48th Annual Symposium on Foundations of Computer Science, FOCS 2007 - Providence, RI, United States Duration: Oct 20 2007 → Oct 23 2007 |

### Other

Other | 48th Annual Symposium on Foundations of Computer Science, FOCS 2007 |
---|---|

Country | United States |

City | Providence, RI |

Period | 10/20/07 → 10/23/07 |

### Fingerprint

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007*(pp. 161-168). [4389489] https://doi.org/10.1109/FOCS.2007.4389489

**Non-linear index coding outperforming the linear optimum.** / Lubetzky, Eyal; Stav, Uri.

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

*Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007.*, 4389489, pp. 161-168, 48th Annual Symposium on Foundations of Computer Science, FOCS 2007, Providence, RI, United States, 10/20/07. https://doi.org/10.1109/FOCS.2007.4389489

}

TY - GEN

T1 - Non-linear index coding outperforming the linear optimum

AU - Lubetzky, Eyal

AU - Stav, Uri

PY - 2007

Y1 - 2007

N2 - The following source coding problem was introduced by Birk and Kol: a sender holds a word x ∈ {0, 1}n, and wishes to broadcast a codeword to n receivers, R1, . . . , Rn. The receiver R1 is interested in xi, and has prior side information comprising some subset of the n bits. This corresponds to a directed graph G on n vertices, where ij is an edge iff Ri knows the bit xj. An index code for G is an encoding scheme which enables each Ri to always reconstruct xi, given his side information. The minimal word length of an index code was studied by Bar-Yossef, Birk, Jayram and Kol [4], They introduced a graph parameter, minrk2(G), which completely characterizes the length of an optimal linear index code for G The authors of [4] 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 [4] in the following strong sense: for any ε > 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 1-ε, and yet a non-linear index code for G has a word length of nε. This is achieved by an explicit construction, which extends Alon's variant of the celebrated Ramsey construction of Frankl and Wilson.

AB - The following source coding problem was introduced by Birk and Kol: a sender holds a word x ∈ {0, 1}n, and wishes to broadcast a codeword to n receivers, R1, . . . , Rn. The receiver R1 is interested in xi, and has prior side information comprising some subset of the n bits. This corresponds to a directed graph G on n vertices, where ij is an edge iff Ri knows the bit xj. An index code for G is an encoding scheme which enables each Ri to always reconstruct xi, given his side information. The minimal word length of an index code was studied by Bar-Yossef, Birk, Jayram and Kol [4], They introduced a graph parameter, minrk2(G), which completely characterizes the length of an optimal linear index code for G The authors of [4] 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 [4] in the following strong sense: for any ε > 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 1-ε, and yet a non-linear index code for G has a word length of nε. This is achieved by an explicit construction, which extends Alon's variant of the celebrated Ramsey construction of Frankl and Wilson.

UR - http://www.scopus.com/inward/record.url?scp=46749126206&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=46749126206&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2007.4389489

DO - 10.1109/FOCS.2007.4389489

M3 - Conference contribution

AN - SCOPUS:46749126206

SN - 0769530109

SN - 9780769530109

SP - 161

EP - 168

BT - Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007

ER -