9 research outputs found
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
Linear Transformations for Randomness Extraction
Information-efficient approaches for extracting randomness from imperfect
sources have been extensively studied, but simpler and faster ones are required
in the high-speed applications of random number generation. In this paper, we
focus on linear constructions, namely, applying linear transformation for
randomness extraction. We show that linear transformations based on sparse
random matrices are asymptotically optimal to extract randomness from
independent sources and bit-fixing sources, and they are efficient (may not be
optimal) to extract randomness from hidden Markov sources. Further study
demonstrates the flexibility of such constructions on source models as well as
their excellent information-preserving capabilities. Since linear
transformations based on sparse random matrices are computationally fast and
can be easy to implement using hardware like FPGAs, they are very attractive in
the high-speed applications. In addition, we explore explicit constructions of
transformation matrices. We show that the generator matrices of primitive BCH
codes are good choices, but linear transformations based on such matrices
require more computational time due to their high densities.Comment: 2 columns, 14 page
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
Linear extractors for extracting randomness from noisy sources
Linear transformations have many applications in information theory, like data compression and error-correcting codes design. In this paper, we study the power of linear transformations in randomness extraction, namely linear extractors, as another important application. Comparing to most existing methods for randomness extraction, linear extractors (especially those constructed with sparse matrices) are computationally fast and can be simply implemented with hardware like FPGAs, which makes them very attractive in practical use. We mainly focus on simple, efficient and sparse constructions of linear extractors. Specifically, we demonstrate that random matrices can generate random bits very efficiently from a variety of noisy sources, including noisy coin sources, bit-fixing sources, noisy (hidden) Markov sources, as well as their mixtures. It shows that low-density random matrices have almost the same efficiency as high-density random matrices when the input sequence is long, which provides a way to simplify hardware/software implementation. Note that although we constructed matrices with randomness, they are deterministic (seedless) extractors - once we constructed them, the same construction can be used for any number of times without using any seeds. Another way to construct linear extractors is based on generator matrices of primitive BCH codes. This method is more explicit, but less practical due to its computational complexity and dimensional constraints
Capacity-achieving Polar-based LDGM Codes
In this paper, we study codes with sparse generator matrices. More
specifically, low-density generator matrix (LDGM) codes with a certain
constraint on the weight of all the columns in the generator matrix are
considered. In this paper, it is first shown that when a binary-input
memoryless symmetric (BMS) channel and a constant are given, there
exists a polarization kernel such that the corresponding polar code is
capacity-achieving and the column weights of the generator matrices are bounded
from above by .
Then, a general construction based on a concatenation of polar codes and a
rate- code, and a new column-splitting algorithm that guarantees a much
sparser generator matrix is given. More specifically, for any BMS channel and
any , where , an existence of
sequence of capacity-achieving codes with all the column wights of the
generator matrix upper bounded by is shown.
Furthermore, coding schemes for BEC and BMS channels, based on a second
column-splitting algorithm are devised with low-complexity decoding that uses
successive-cancellation. The second splitting algorithm allows for the use of a
low-complexity decoder by preserving the reliability of the bit-channels
observed by the source bits, and by increasing the code block length. In
particular, for any BEC and any , the
existence of a sequence of capacity-achieving codes where all the column wights
of the generator matrix are bounded from above by and
with decoding complexity is shown. The existence of similar
capacity-achieving LDGM codes with low-complexity decoding is shown for any BMS
channel, and for any .Comment: arXiv admin note: text overlap with arXiv:2001.1198
From Polar to Reed-Muller Codes:Unified Scaling, Non-standard Channels, and a Proven Conjecture
The year 2016, in which I am writing these words, marks the centenary of Claude Shannon, the father of information theory. In his landmark 1948 paper "A Mathematical Theory of Communication", Shannon established the largest rate at which reliable communication is possible, and he referred to it as the channel capacity. Since then, researchers have focused on the design of practical coding schemes that could approach such a limit. The road to channel capacity has been almost 70 years long and, after many ideas, occasional detours, and some rediscoveries, it has culminated in the description of low-complexity and provably capacity-achieving coding schemes, namely, polar codes and iterative codes based on sparse graphs. However, next-generation communication systems require an unprecedented performance improvement and the number of transmission settings relevant in applications is rapidly increasing. Hence, although Shannon's limit seems finally close at hand, new challenges are just around the corner. In this thesis, we trace a road that goes from polar to Reed-Muller codes and, by doing so, we investigate three main topics: unified scaling, non-standard channels, and capacity via symmetry. First, we consider unified scaling. A coding scheme is capacity-achieving when, for any rate smaller than capacity, the error probability tends to 0 as the block length becomes increasingly larger. However, the practitioner is often interested in more specific questions such as, "How much do we need to increase the block length in order to halve the gap between rate and capacity?". We focus our analysis on polar codes and develop a unified framework to rigorously analyze the scaling of the main parameters, i.e., block length, rate, error probability, and channel quality. Furthermore, in light of the recent success of a list decoding algorithm for polar codes, we provide scaling results on the performance of list decoders. Next, we deal with non-standard channels. When we say that a coding scheme achieves capacity, we typically consider binary memoryless symmetric channels. However, practical transmission scenarios often involve more complicated settings. For example, the downlink of a cellular system is modeled as a broadcast channel, and the communication on fiber links is inherently asymmetric. We propose provably optimal low-complexity solutions for these settings. In particular, we present a polar coding scheme that achieves the best known rate region for the broadcast channel, and we describe three paradigms to achieve the capacity of asymmetric channels. To do so, we develop general coding "primitives", such as the chaining construction that has already proved to be useful in a variety of communication problems. Finally, we show how to achieve capacity via symmetry. In the early days of coding theory, a popular paradigm consisted in exploiting the structure of algebraic codes to devise practical decoding algorithms. However, proving the optimality of such coding schemes remained an elusive goal. In particular, the conjecture that Reed-Muller codes achieve capacity dates back to the 1960s. We solve this open problem by showing that Reed-Muller codes and, in general, codes with sufficient symmetry are capacity-achieving over erasure channels under optimal MAP decoding. As the proof does not rely on the precise structure of the codes, we are able to show that symmetry alone guarantees optimal performance
Polarization and randomness extraction
This paper explores a connection between randomness extraction and channel (source) coding problems. It is explained how efficient extractors can be used to define efficient coding schemes and reciprocally, a new deterministic extractor based on a polar coding scheme is proposed. Since the source model used in extractors for computer science (cryptography) do not assume i.i.d. or known distributions, a generalized polarization phenomenon for sources with block memory and unknown distributions is developed. It is shown that in this setting, the min-entropy (as usual in extractors) rather than Shannon entropy can be efficiently extracted. The derived polar coding results also apply to compound channels with memory