Privileged users in zero-error transmission over a noisy channel

The k-th power of a graph G is the graph whose vertex set is V (G)k , where two distinct ktuples are adjacent iff they are equal or adjacent in G in each coordinate. The Shannon capacity of G, c(G), is limk→∞ α(Gk ) 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 Ci, describes a setting when there are t ≥ 2 senders, each with his own channel Ci, 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 Gi. It is well known that c(G) ≥ i c(Gi), 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 Ci to each sender i ∈ [t], such that the capacity of a group of senders X ⊂ [t] is h...

• Type: Journal
• Year: 2007
• Where: COMBINATORICA
• Authors: Noga Alon, Eyal Lubetzky
• Keywords: Characteristic Graphs, COMBINATORICA 2007, Effective Alphabet Size, K-th Power