### Abstract

The k-th power of a graph G is the graph whose vertex set is V(G) ^{k} , where two distinct k-tuples are adjacent iff they are equal or adjacent in G in each coordinate. The Shannon capacity of G, c(G), is lim _{k→∞} α(G ^{k} )^{1/k} , where α(G) denotes the independence number of G. When G is the characteristic graph of a channel C, c(G) measures the effective alphabet size of C in a zero-error protocol. A sum of channels, C = ∑ _{i} C _{i} , describes a setting when there are t ≥ 2 senders, each with his own channel C _{i} , and each letter in a word can be selected from any of the channels. This corresponds to a disjoint union of the characteristic graphs, G = ∑ _{i} G _{i} . It is well known that c(G) ≥ ∑ _{i} c(G _{i} ), and in [1] it is shown that in fact c(G) can be larger than any fixed power of the above sum. We extend the ideas of [1] and show that for every F, a family of subsets of [t], it is possible to assign a channel C _{i} to each sender i [t], such that the capacity of a group of senders X ⊂ [t] is high iff X contains some F F. This corresponds to a case where only privileged subsets of senders are allowed to transmit in a high rate. For instance, as an analogue to secret sharing, it is possible to ensure that whenever at least k senders combine their channels, they obtain a high capacity, however every group of k - 1 senders has a low capacity (and yet is not totally denied of service). In the process, we obtain an explicit Ramsey construction of an edge-coloring of the complete graph on n vertices by t colors, where every induced subgraph on exp (Ω (√log n\log \log n}) vertices contains all t colors.

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

Pages (from-to) | 737-743 |

Number of pages | 7 |

Journal | Combinatorica |

Volume | 27 |

Issue number | 6 |

DOIs | |

State | Published - Nov 2007 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Combinatorica*,

*27*(6), 737-743. https://doi.org/10.1007/s00493-007-2263-z

**Privileged users in zero-error transmission over a noisy channel.** / Alon, Noga; Lubetzky, Eyal.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 27, no. 6, pp. 737-743. https://doi.org/10.1007/s00493-007-2263-z

}

TY - JOUR

T1 - Privileged users in zero-error transmission over a noisy channel

AU - Alon, Noga

AU - Lubetzky, Eyal

PY - 2007/11

Y1 - 2007/11

N2 - The k-th power of a graph G is the graph whose vertex set is V(G) k , where two distinct k-tuples are adjacent iff they are equal or adjacent in G in each coordinate. The Shannon capacity of G, c(G), is lim k→∞ α(G k )1/k , where α(G) denotes the independence number of G. When G is the characteristic graph of a channel C, c(G) measures the effective alphabet size of C in a zero-error protocol. A sum of channels, C = ∑ i C i , describes a setting when there are t ≥ 2 senders, each with his own channel C i , and each letter in a word can be selected from any of the channels. This corresponds to a disjoint union of the characteristic graphs, G = ∑ i G i . It is well known that c(G) ≥ ∑ i c(G i ), and in [1] it is shown that in fact c(G) can be larger than any fixed power of the above sum. We extend the ideas of [1] and show that for every F, a family of subsets of [t], it is possible to assign a channel C i to each sender i [t], such that the capacity of a group of senders X ⊂ [t] is high iff X contains some F F. This corresponds to a case where only privileged subsets of senders are allowed to transmit in a high rate. For instance, as an analogue to secret sharing, it is possible to ensure that whenever at least k senders combine their channels, they obtain a high capacity, however every group of k - 1 senders has a low capacity (and yet is not totally denied of service). In the process, we obtain an explicit Ramsey construction of an edge-coloring of the complete graph on n vertices by t colors, where every induced subgraph on exp (Ω (√log n\log \log n}) vertices contains all t colors.

AB - The k-th power of a graph G is the graph whose vertex set is V(G) k , where two distinct k-tuples are adjacent iff they are equal or adjacent in G in each coordinate. The Shannon capacity of G, c(G), is lim k→∞ α(G k )1/k , where α(G) denotes the independence number of G. When G is the characteristic graph of a channel C, c(G) measures the effective alphabet size of C in a zero-error protocol. A sum of channels, C = ∑ i C i , describes a setting when there are t ≥ 2 senders, each with his own channel C i , and each letter in a word can be selected from any of the channels. This corresponds to a disjoint union of the characteristic graphs, G = ∑ i G i . It is well known that c(G) ≥ ∑ i c(G i ), and in [1] it is shown that in fact c(G) can be larger than any fixed power of the above sum. We extend the ideas of [1] and show that for every F, a family of subsets of [t], it is possible to assign a channel C i to each sender i [t], such that the capacity of a group of senders X ⊂ [t] is high iff X contains some F F. This corresponds to a case where only privileged subsets of senders are allowed to transmit in a high rate. For instance, as an analogue to secret sharing, it is possible to ensure that whenever at least k senders combine their channels, they obtain a high capacity, however every group of k - 1 senders has a low capacity (and yet is not totally denied of service). In the process, we obtain an explicit Ramsey construction of an edge-coloring of the complete graph on n vertices by t colors, where every induced subgraph on exp (Ω (√log n\log \log n}) vertices contains all t colors.

UR - http://www.scopus.com/inward/record.url?scp=44649100396&partnerID=8YFLogxK

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U2 - 10.1007/s00493-007-2263-z

DO - 10.1007/s00493-007-2263-z

M3 - Article

AN - SCOPUS:44649100396

VL - 27

SP - 737

EP - 743

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 6

ER -