33 research outputs found
A Characterization of the Shannon Ordering of Communication Channels
The ordering of communication channels was first introduced by Shannon. In
this paper, we aim to find a characterization of the Shannon ordering. We show
that contains if and only if is the skew-composition of with
a convex-product channel. This fact is used to derive a characterization of the
Shannon ordering that is similar to the Blackwell-Sherman-Stein theorem. Two
channels are said to be Shannon-equivalent if each one is contained in the
other. We investigate the topologies that can be constructed on the space of
Shannon-equivalent channels. We introduce the strong topology and the BRM
metric on this space. Finally, we study the continuity of a few channel
parameters and operations under the strong topology.Comment: 23 pages, presented in part at ISIT'17. arXiv admin note: text
overlap with arXiv:1702.0072
Topological Structures on DMC spaces
Two channels are said to be equivalent if they are degraded from each other.
The space of equivalent channels with input alphabet and output alphabet
can be naturally endowed with the quotient of the Euclidean topology by the
equivalence relation. A topology on the space of equivalent channels with fixed
input alphabet and arbitrary but finite output alphabet is said to be
natural if and only if it induces the quotient topology on the subspaces of
equivalent channels sharing the same output alphabet. We show that every
natural topology is -compact, separable and path-connected. On the
other hand, if , a Hausdorff natural topology is not Baire and it is
not locally compact anywhere. This implies that no natural topology can be
completely metrized if . The finest natural topology, which we call
the strong topology, is shown to be compactly generated, sequential and .
On the other hand, the strong topology is not first-countable anywhere, hence
it is not metrizable. We show that in the strong topology, a subspace is
compact if and only if it is rank-bounded and strongly-closed. We introduce a
metric distance on the space of equivalent channels which compares the noise
levels between channels. The induced metric topology, which we call the
noisiness topology, is shown to be natural. We also study topologies that are
inherited from the space of meta-probability measures by identifying channels
with their Blackwell measures. We show that the weak-* topology is exactly the
same as the noisiness topology and hence it is natural. We prove that if
, the total variation topology is not natural nor Baire, hence it is
not completely metrizable. Moreover, it is not locally compact anywhere.
Finally, we show that the Borel -algebra is the same for all Hausdorff
natural topologies.Comment: 43 pages, submitted to IEEE Trans. Inform. Theory and in part to
ISIT201
Continuity of Channel Parameters and Operations under Various DMC Topologies
We study the continuity of many channel parameters and operations under
various topologies on the space of equivalent discrete memoryless channels
(DMC). We show that mutual information, channel capacity, Bhattacharyya
parameter, probability of error of a fixed code, and optimal probability of
error for a given code rate and blocklength, are continuous under various DMC
topologies. We also show that channel operations such as sums, products,
interpolations, and Ar{\i}kan-style transformations are continuous.Comment: 31 pages. Submitted to IEEE Trans. Inform. Theory and in part to
ISIT201
On the Input-Degradedness and Input-Equivalence Between Channels
A channel is said to be input-degraded from another channel if
can be simulated from by randomization at the input. We provide a
necessary and sufficient condition for a channel to be input-degraded from
another one. We show that any decoder that is good for is also good for
. We provide two characterizations for input-degradedness, one of which is
similar to the Blackwell-Sherman-Stein theorem. We say that two channels are
input-equivalent if they are input-degraded from each other. We study the
topologies that can be constructed on the space of input-equivalent channels,
and we investigate their properties. Moreover, we study the continuity of
several channel parameters and operations under these topologies.Comment: 30 pages. Submitted to IEEE Trans. Inform. Theory and in part to
ISIT2017. arXiv admin note: substantial text overlap with arXiv:1701.0446
Oriented paths in n-chromatic digraphs
In this thesis, we try to treat the problem of oriented paths in n-chromatic
digraphs. We first treat the case of antidirected paths in 5-chromatic
digraphs, where we explain El-Sahili's theorem and provide an elementary and
shorter proof of it. We then treat the case of paths with two blocks in
n-chromatic digraphs with n greater than 4, where we explain the two different
approaches of Addario-Berry et al. and of El-Sahili. We indicate a mistake in
Addario-Berry et al.'s proof and provide a correction for it.Comment: 25 pages, Master thesis in Graph Theory at the Lebanese Universit
Ergodic Theory Meets Polarization. I: An Ergodic Theory for Binary Operations
An open problem in polarization theory is to determine the binary operations
that always lead to polarization (in the general multilevel sense) when they
are used in Ar{\i}kan style constructions. This paper, which is presented in
two parts, solves this problem by providing a necessary and sufficient
condition for a binary operation to be polarizing. This (first) part of the
paper introduces the mathematical framework that we will use in the second part
to characterize the polarizing operations. We define uniformity preserving,
irreducible, ergodic and strongly ergodic operations and we study their
properties. The concepts of a stable partition and the residue of a stable
partition are introduced. We show that an ergodic operation is strongly ergodic
if and only if all its stable partitions are their own residues. We also study
the products of binary operations and the structure of their stable partitions.
We show that the product of a sequence of binary operations is strongly ergodic
if and only if all the operations in the sequence are strongly ergodic. In the
second part of the paper, we provide a foundation of polarization theory based
on the ergodic theory of binary operations that we develop in this part.Comment: 34 pages, 1 figure. Accepted to IEEE Trans. Inform. Theory and
presented in part at ISIT'1
Polar Codes for Arbitrary DMCs and Arbitrary MACs
Polar codes are constructed for arbitrary channels by imposing an arbitrary
quasigroup structure on the input alphabet. Just as with "usual" polar codes,
the block error probability under successive cancellation decoding is
, where is the block length. Encoding and
decoding for these codes can be implemented with a complexity of .
It is shown that the same technique can be used to construct polar codes for
arbitrary multiple access channels (MAC) by using an appropriate Abelian group
structure. Although the symmetric sum capacity is achieved by this coding
scheme, some points in the symmetric capacity region may not be achieved. In
the case where the channel is a combination of linear channels, we provide a
necessary and sufficient condition characterizing the channels whose symmetric
capacity region is preserved by the polarization process. We also provide a
sufficient condition for having a maximal loss in the dominant face.Comment: 32 pages, 1 figure. arXiv admin note: text overlap with
arXiv:1112.177
Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs
We prove polarization theorems for arbitrary classical-quantum (cq) channels.
The input alphabet is endowed with an arbitrary Abelian group operation and an
Ar{\i}kan-style transformation is applied using this operation. It is shown
that as the number of polarization steps becomes large, the synthetic
cq-channels polarize to deterministic homomorphism channels which project their
input to a quotient group of the input alphabet. This result is used to
construct polar codes for arbitrary cq-channels and arbitrary classical-quantum
multiple access channels (cq-MAC). The encoder can be implemented in operations, where is the blocklength of the code. A quantum successive
cancellation decoder for the constructed codes is proposed. It is shown that
the probability of error of this decoder decays faster than
for any .Comment: 30 pages. Submitted to IEEE Trans. Inform. Theory and in part to
ISIT201