66 research outputs found
Polar Coding for Achieving the Capacity of Marginal Channels in Nonbinary-Input Setting
Achieving information-theoretic security using explicit coding scheme in
which unlimited computational power for eavesdropper is assumed, is one of the
main topics is security consideration. It is shown that polar codes are
capacity achieving codes and have a low complexity in encoding and decoding. It
has been proven that polar codes reach to secrecy capacity in the binary-input
wiretap channels in symmetric settings for which the wiretapper's channel is
degraded with respect to the main channel. The first task of this paper is to
propose a coding scheme to achieve secrecy capacity in asymmetric
nonbinary-input channels while keeping reliability and security conditions
satisfied. Our assumption is that the wiretap channel is stochastically
degraded with respect to the main channel and message distribution is
unspecified. The main idea is to send information set over good channels for
Bob and bad channels for Eve and send random symbols for channels that are good
for both. In this scheme the frozen vector is defined over all possible choices
using polar codes ensemble concept. We proved that there exists a frozen vector
for which the coding scheme satisfies reliability and security conditions. It
is further shown that uniform distribution of the message is the necessary
condition for achieving secrecy capacity.Comment: Accepted to be published in "51th Conference on Information Sciences
and Systems", Baltimore, Marylan
On the Construction of Polar Codes for Achieving the Capacity of Marginal Channels
Achieving security against adversaries with unlimited computational power is
of great interest in a communication scenario. Since polar codes are capacity
achieving codes with low encoding-decoding complexity and they can approach
perfect secrecy rates for binary-input degraded wiretap channels in symmetric
settings, they are investigated extensively in the literature recently. In this
paper, a polar coding scheme to achieve secrecy capacity in non-symmetric
binary input channels is proposed. The proposed scheme satisfies security and
reliability conditions. The wiretap channel is assumed to be stochastically
degraded with respect to the legitimate channel and message distribution is
uniform. The information set is sent over channels that are good for Bob and
bad for Eve. Random bits are sent over channels that are good for both Bob and
Eve. A frozen vector is chosen randomly and is sent over channels bad for both.
We prove that there exists a frozen vector for which the coding scheme
satisfies reliability and security conditions and approaches the secrecy
capacity. We further empirically show that in the proposed scheme for
non-symmetric binary-input discrete memoryless channels, the equivocation rate
achieves its upper bound in the whole capacity-equivocation region
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
Design and Analysis of Communication Schemes via Polar Coding
Polar codes, introduced by Arikan in 2009, gave the first solution to the problem of designing explicit coding schemes that attain Shannon capacity of several basic models of communication channels. This discovery made it possible to attain theoretical limits of communication in a number of other problems of data compression and multi-user communication as well as provided new perspectives on extremal configurations of some discrete-time random walks.
This thesis is devoted to the design of communication protocols for several basic information-theoretic problems as well to the problem of efficient construction of polar codes.
In the first part we consider the problem of optimizing the amount of data transmit- ted between two terminals performing interactive computation of a function. Information- theoretic limits for one model of interactive computation were found in recent literature. We consider the distributed source coding problem that arises in the analysis of this model, designing a polar coding scheme that serves the basis for the distributed computation. As a result, it becomes possible to attain the smallest possible rate of data exchange between the terminals using an explicit protocol of encoding and data exchange that supports reli- able computation of the function by both parties. We also extend our considerations to a multi-terminal variation of this problem.
Secondly, we turn to the problem of communication between two parties over a link observed by an adversary, known as the âwiretap channel.â Explicit capacity-achieving schemes for various models of the wiretap channel have received significant attention in recent literature. In this work, we address the general model of the channel, removing the constraints on the channels adopted in the earlier works. We show that secrecy capacity of the wiretap channel under a âstrong secrecy constraintâ can be achieved using an ex- plicit scheme based on polar codes. We also extend our construction to the case of the broadcast channel with confidential messages due to Csiszar and Korner, achieving the entire capacity region of this communication model.
In the last part of the thesis we consider the problem of efficient construction of polar codes. While Arıkanâs scheme is explicit, his original proposal suffers from high construction complexity which grows exponentially with the number of evolution steps. An approximation procedure for binary-input channels was proposed and analyzed in the literature. Here we propose and study a construction algorithm for polar codes with arbitrarily-sized input alphabets. We establish a complexity estimate of the algorithm and derive an estimate of the approximation error that ensues from its use. The approximation error reduces the gap to the recently established lower bound for this type of algorithms. The validity of the proposed algorithm is supported by experimental results
Construction of Capacity-Achieving Lattice Codes: Polar Lattices
In this paper, we propose a new class of lattices constructed from polar
codes, namely polar lattices, to achieve the capacity \frac{1}{2}\log(1+\SNR)
of the additive white Gaussian-noise (AWGN) channel. Our construction follows
the multilevel approach of Forney \textit{et al.}, where we construct a
capacity-achieving polar code on each level. The component polar codes are
shown to be naturally nested, thereby fulfilling the requirement of the
multilevel lattice construction. We prove that polar lattices are
\emph{AWGN-good}. Furthermore, using the technique of source polarization, we
propose discrete Gaussian shaping over the polar lattice to satisfy the power
constraint. Both the construction and shaping are explicit, and the overall
complexity of encoding and decoding is for any fixed target error
probability.Comment: full version of the paper to appear in IEEE Trans. Communication
Irregular polar coding for complexity-constrained lightwave systems
Next-generation fiber-optic communications call for ultra-reliable forward error correction codes that are capable of low-power and low-latency decoding. In this paper, we propose a new class of polar codes, whose polarization units are irregularly pruned to reduce computational complexity and decoding latency without sacrificing error correction performance. We then experimentally demonstrate that the proposed irregular polar codes can outperform state-of-the-art low-density parity-check (LDPC) codes, while decoding complexity and latency can be reduced by at least 30% and 70%, respectively, versus regular polar codes, while also obtaining a marginal performance improvement
Construction of lattices for communications and security
In this thesis, we propose a new class of lattices based on polar codes, namely polar lattices. Polar lattices enjoy explicit construction and provable goodness for the additive white Gaussian noise (AWGN) channel, \textit{i.e.}, they are \emph{AWGN-good} lattices, in the sense that the error probability (for infinite lattice coding) vanishes for any fixed volume-to-noise ratio (VNR) greater than . Our construction is based on the multilevel approach of Forney \textit{et al.}, where on each level we construct a capacity-achieving polar code. We show the component polar codes are naturally nested, thereby fulfilling the requirement of the multilevel lattice construction. We present a more precise analysis of the VNR of the resultant lattice, which is upper-bounded in terms of the flatness factor and the capacity losses of the component codes. The proposed polar lattices are efficiently decodable by using multi-stage decoding. Design examples are presented to demonstrate the superior performance of polar lattices.
However, there is no infinite lattice coding in the practical applications. We need to apply the power constraint on the polar lattices which generates the polar lattice codes. We prove polar lattice codes can achieve the capacity \frac{1}{2}\log(1+\SNR) of the power-constrained AWGN channel with a novel shaping scheme. The main idea is that by implementing the lattice Gaussian distribution over the AWGN-good polar lattices, the maximum error-free transmission rate of the resultant coding scheme can be arbitrarily close to the capacity \frac{1}{2}\log(1+\SNR). The shaping technique is based on discrete lattice Gaussian distribution, which leads to a binary asymmetric channel at each level for the multilevel lattice codes. Then it is straightforward to employ multilevel asymmetric polar codes which is a combination of polar lossless source coding and polar channel coding. The construction of polar codes for an asymmetric channel can be converted to that for a related symmetric channel, and it turns out that this symmetric channel is equivalent to an minimum mean-square error (MMSE) scaled channel in lattice coding in terms of polarization, which eventually simplifies our coding design.
Finally, we investigate the application of polar lattices in physical layer security. Polar lattice codes are proved to be able to achieve the strong secrecy capacity of the Mod- AWGN wiretap channel. The Mod- assumption was due to the fact that a practical shaping scheme aiming to achieve the optimum shaping gain was missing. In this thesis, we use our shaping scheme and extend polar lattice coding to the Gaussian wiretap channel. By employing the polar coding technique for asymmetric channels, we manage to construct an AWGN-good lattice and a secrecy-good lattice with optimal shaping simultaneously. Then we prove the resultant wiretap coding scheme can achieve the strong secrecy capacity for the Gaussian wiretap channel.Open Acces
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