29 research outputs found
On Index Coding and Graph Homomorphism
In this work, we study the problem of index coding from graph homomorphism
perspective. We show that the minimum broadcast rate of an index coding problem
for different variations of the problem such as non-linear, scalar, and vector
index code, can be upper bounded by the minimum broadcast rate of another index
coding problem when there exists a homomorphism from the complement of the side
information graph of the first problem to that of the second problem. As a
result, we show that several upper bounds on scalar and vector index code
problem are special cases of one of our main theorems.
For the linear scalar index coding problem, it has been shown in [1] that the
binary linear index of a graph is equal to a graph theoretical parameter called
minrank of the graph. For undirected graphs, in [2] it is shown that
if and only if there exists a homomorphism from
to a predefined graph . Combining these two results, it
follows that for undirected graphs, all the digraphs with linear index of at
most k coincide with the graphs for which there exists a homomorphism from
to . In this paper, we give a direct proof to this result
that works for digraphs as well.
We show how to use this classification result to generate lower bounds on
scalar and vector index. In particular, we provide a lower bound for the scalar
index of a digraph in terms of the chromatic number of its complement.
Using our framework, we show that by changing the field size, linear index of
a digraph can be at most increased by a factor that is independent from the
number of the nodes.Comment: 5 pages, to appear in "IEEE Information Theory Workshop", 201
On AVCs with Quadratic Constraints
In this work we study an Arbitrarily Varying Channel (AVC) with quadratic
power constraints on the transmitter and a so-called "oblivious" jammer (along
with additional AWGN) under a maximum probability of error criterion, and no
private randomness between the transmitter and the receiver. This is in
contrast to similar AVC models under the average probability of error criterion
considered in [1], and models wherein common randomness is allowed [2] -- these
distinctions are important in some communication scenarios outlined below.
We consider the regime where the jammer's power constraint is smaller than
the transmitter's power constraint (in the other regime it is known no positive
rate is possible). For this regime we show the existence of stochastic codes
(with no common randomness between the transmitter and receiver) that enables
reliable communication at the same rate as when the jammer is replaced with
AWGN with the same power constraint. This matches known information-theoretic
outer bounds. In addition to being a stronger result than that in [1] (enabling
recovery of the results therein), our proof techniques are also somewhat more
direct, and hence may be of independent interest.Comment: A shorter version of this work will be send to ISIT13, Istanbul. 8
pages, 3 figure