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

Noga Alon, Eyal Lubetzky

Research output: Contribution to journalArticle

### 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) 737-743 7 Combinatorica 27 6 https://doi.org/10.1007/s00493-007-2263-z Published - Nov 2007

### Fingerprint

Color
Zero
Coloring
Graph in graph theory
Secret Sharing
Subset
Independence number
Edge Coloring
Induced Subgraph
Complete Graph
Assign
Disjoint
Union
Denote
Analogue
Distinct
Vertex of a graph
Family

### ASJC Scopus subject areas

• Mathematics(all)
• Discrete Mathematics and Combinatorics

### Cite this

In: Combinatorica, Vol. 27, No. 6, 11.2007, p. 737-743.

Research output: Contribution to journalArticle

@article{a296c8c3fa434cf7a6ae10fc40362a8b,
title = "Privileged users in zero-error transmission over a noisy channel",
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.",
author = "Noga Alon and Eyal Lubetzky",
year = "2007",
month = "11",
doi = "10.1007/s00493-007-2263-z",
language = "English (US)",
volume = "27",
pages = "737--743",
journal = "Combinatorica",
issn = "0209-9683",
publisher = "Janos Bolyai Mathematical Society",
number = "6",

}

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

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

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 -