34 research outputs found
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
Universal Polar Codes for More Capable and Less Noisy Channels and Sources
We prove two results on the universality of polar codes for source coding and
channel communication. First, we show that for any polar code built for a
source there exists a slightly modified polar code - having the same
rate, the same encoding and decoding complexity and the same error rate - that
is universal for every source when using successive cancellation
decoding, at least when the channel is more capable than
and is such that it maximizes for the given channels
and . This result extends to channel coding for discrete
memoryless channels. Second, we prove that polar codes using successive
cancellation decoding are universal for less noisy discrete memoryless
channels.Comment: 10 pages, 3 figure
Asymptotic Distribution of Multilevel Channel Polarization for a Certain Class of Erasure Channels
This study examines multilevel channel polarization for a certain class of
erasure channels that the input alphabet size is an arbitrary composite number.
We derive limiting proportions of partially noiseless channels for such a
class. The results of this study are proved by an argument of convergent
sequences, inspired by Alsan and Telatar's simple proof of polarization, and
without martingale convergence theorems for polarization process.Comment: 31 pages; 1 figure; 1 table; a short version of this paper has been
submitted to the 2018 IEEE International Symposium on Information Theory
(ISIT2018
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
How to Achieve the Capacity of Asymmetric Channels
We survey coding techniques that enable reliable transmission at rates that
approach the capacity of an arbitrary discrete memoryless channel. In
particular, we take the point of view of modern coding theory and discuss how
recent advances in coding for symmetric channels help provide more efficient
solutions for the asymmetric case. We consider, in more detail, three basic
coding paradigms.
The first one is Gallager's scheme that consists of concatenating a linear
code with a non-linear mapping so that the input distribution can be
appropriately shaped. We explicitly show that both polar codes and spatially
coupled codes can be employed in this scenario. Furthermore, we derive a
scaling law between the gap to capacity, the cardinality of the input and
output alphabets, and the required size of the mapper.
The second one is an integrated scheme in which the code is used both for
source coding, in order to create codewords distributed according to the
capacity-achieving input distribution, and for channel coding, in order to
provide error protection. Such a technique has been recently introduced by
Honda and Yamamoto in the context of polar codes, and we show how to apply it
also to the design of sparse graph codes.
The third paradigm is based on an idea of B\"ocherer and Mathar, and
separates the two tasks of source coding and channel coding by a chaining
construction that binds together several codewords. We present conditions for
the source code and the channel code, and we describe how to combine any source
code with any channel code that fulfill those conditions, in order to provide
capacity-achieving schemes for asymmetric channels. In particular, we show that
polar codes, spatially coupled codes, and homophonic codes are suitable as
basic building blocks of the proposed coding strategy.Comment: 32 pages, 4 figures, presented in part at Allerton'14 and published
in IEEE Trans. Inform. Theor
Achieving the Capacity of any DMC using only Polar Codes
We construct a channel coding scheme to achieve the capacity of any discrete
memoryless channel based solely on the techniques of polar coding. In
particular, we show how source polarization and randomness extraction via
polarization can be employed to "shape" uniformly-distributed i.i.d. random
variables into approximate i.i.d. random variables distributed ac- cording to
the capacity-achieving distribution. We then combine this shaper with a variant
of polar channel coding, constructed by the duality with source coding, to
achieve the channel capacity. Our scheme inherits the low complexity encoder
and decoder of polar coding. It differs conceptually from Gallager's method for
achieving capacity, and we discuss the advantages and disadvantages of the two
schemes. An application to the AWGN channel is discussed.Comment: 9 pages, 7 figure