19 research outputs found
On secure network coding with uniform wiretap sets
This paper shows determining the secrecy capacity of a unicast network with
uniform wiretap sets is at least as difficult as the k-unicast problem. In
particular, we show that a general k-unicast problem can be reduced to the
problem of finding the secrecy capacity of a corresponding single unicast
network with uniform link capacities and one arbitrary wiretap link
General Scheme for Perfect Quantum Network Coding with Free Classical Communication
This paper considers the problem of efficiently transmitting quantum states
through a network. It has been known for some time that without additional
assumptions it is impossible to achieve this task perfectly in general --
indeed, it is impossible even for the simple butterfly network. As additional
resource we allow free classical communication between any pair of network
nodes. It is shown that perfect quantum network coding is achievable in this
model whenever classical network coding is possible over the same network when
replacing all quantum capacities by classical capacities. More precisely, it is
proved that perfect quantum network coding using free classical communication
is possible over a network with source-target pairs if there exists a
classical linear (or even vector linear) coding scheme over a finite ring. Our
proof is constructive in that we give explicit quantum coding operations for
each network node. This paper also gives an upper bound on the number of
classical communication required in terms of , the maximal fan-in of any
network node, and the size of the network.Comment: 12 pages, 2 figures, generalizes some of the results in
arXiv:0902.1299 to the k-pair problem and codes over rings. Appeared in the
Proceedings of the 36th International Colloquium on Automata, Languages and
Programming (ICALP'09), LNCS 5555, pp. 622-633, 200
On network coding for sum-networks
A directed acyclic network is considered where all the terminals need to
recover the sum of the symbols generated at all the sources. We call such a
network a sum-network. It is shown that there exists a solvably (and linear
solvably) equivalent sum-network for any multiple-unicast network, and thus for
any directed acyclic communication network. It is also shown that there exists
a linear solvably equivalent multiple-unicast network for every sum-network. It
is shown that for any set of polynomials having integer coefficients, there
exists a sum-network which is scalar linear solvable over a finite field F if
and only if the polynomials have a common root in F. For any finite or cofinite
set of prime numbers, a network is constructed which has a vector linear
solution of any length if and only if the characteristic of the alphabet field
is in the given set. The insufficiency of linear network coding and
unachievability of the network coding capacity are proved for sum-networks by
using similar known results for communication networks. Under fractional vector
linear network coding, a sum-network and its reverse network are shown to be
equivalent. However, under non-linear coding, it is shown that there exists a
solvable sum-network whose reverse network is not solvable.Comment: Accepted to IEEE Transactions on Information Theor
An Equivalence Between Secure Network and Index Coding
We extend the equivalence between network coding and index coding by Effros,
El Rouayheb, and Langberg to the secure communication setting in the presence
of an eavesdropper. Specifically, we show that the most general versions of
secure network-coding setup by Chan and Grant and the secure index-coding setup
by Dau, Skachek, and Chee, which also include the randomised encoding setting,
are equivalent
Linear Network Coding for Two-Unicast- Networks: A Commutative Algebraic Perspective and Fundamental Limits
We consider a two-unicast- network over a directed acyclic graph of unit
capacitated edges; the two-unicast- network is a special case of two-unicast
networks where one of the destinations has apriori side information of the
unwanted (interfering) message. In this paper, we settle open questions on the
limits of network coding for two-unicast- networks by showing that the
generalized network sharing bound is not tight, vector linear codes outperform
scalar linear codes, and non-linear codes outperform linear codes in general.
We also develop a commutative algebraic approach to deriving linear network
coding achievability results, and demonstrate our approach by providing an
alternate proof to the previous results of C. Wang et. al., I. Wang et. al. and
Shenvi et. al. regarding feasibility of rate in the network.Comment: A short version of this paper is published in the Proceedings of The
IEEE International Symposium on Information Theory (ISIT), June 201
Guessing Numbers of Odd Cycles
For a given number of colours, , the guessing number of a graph is the
base logarithm of the size of the largest family of colourings of the
vertex set of the graph such that the colour of each vertex can be determined
from the colours of the vertices in its neighbourhood. An upper bound for the
guessing number of the -vertex cycle graph is . It is known that
the guessing number equals whenever is even or is a perfect
square \cite{Christofides2011guessing}. We show that, for any given integer
, if is the largest factor of less than or equal to
, for sufficiently large odd , the guessing number of with
colours is . This answers a question posed by
Christofides and Markstr\"{o}m in 2011 \cite{Christofides2011guessing}. We also
present an explicit protocol which achieves this bound for every . Linking
this to index coding with side information, we deduce that the information
defect of with colours is for sufficiently
large odd . Our results are a generalisation of the case which was
proven in \cite{bar2011index}.Comment: 16 page