 
Summary: Privileged users in zeroerror transmission over a noisy channel
Noga Alon
Eyal Lubetzky
January 11, 2007
Abstract
The kth 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 limk (Gk
)
1
k , where (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
zeroerror 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
