Privileged users in zero-error transmission over a noisy channel

Noga Alon, Eyal Lubetzky

Research output: Contribution to journalArticlepeer-review

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

ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics
• Computational Mathematics