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Privileged users in zero-error transmission over a noisy channel Eyal Lubetzky

Summary: Privileged users in zero-error transmission over a noisy channel
Noga Alon
Eyal Lubetzky
January 11, 2007
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 limk (Gk
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
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


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics