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 = Σ 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 = Σ i G i . It is well known that c(G) ≥ Σ 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 FF. 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 \( (\Omega (\sqrt {\log n\log \log n} )) \) vertices contains all t colors.

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Correspondence to Noga Alon.

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Research supported in part by a USA-Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.

Research partially supported by a Charles Clore Foundation Fellowship.

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Alon, N., Lubetzky, E. Privileged users in zero-error transmission over a noisy channel. Combinatorica 27, 737–743 (2007). https://doi.org/10.1007/s00493-007-2263-z

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Mathematics Subject Classification (2000)

  • 05C35
  • 05C55
  • 94A24