13 research outputs found
Achieving the Uniform Rate Region of General Multiple Access Channels by Polar Coding
We consider the problem of polar coding for transmission over -user
multiple access channels. In the proposed scheme, all users encode their
messages using a polar encoder, while a multi-user successive cancellation
decoder is deployed at the receiver. The encoding is done separately across the
users and is independent of the target achievable rate. For the code
construction, the positions of information bits and frozen bits for each of the
users are decided jointly. This is done by treating the polar transformations
across all the users as a single polar transformation with a certain
\emph{polarization base}. We characterize the resolution of achievable rates on
the dominant face of the uniform rate region in terms of the number of users
and the length of the polarization base . In particular, we prove that
for any target rate on the dominant face, there exists an achievable rate, also
on the dominant face, within the distance at most
from the target rate. We then prove that the proposed MAC polar coding scheme
achieves the whole uniform rate region with fine enough resolution by changing
the decoding order in the multi-user successive cancellation decoder, as
and the code block length grow large. The encoding and decoding
complexities are and the asymptotic block error probability of
is guaranteed. Examples of achievable rates for
the -user multiple access channel are provided
Polar codes in network quantum information theory
Polar coding is a method for communication over noisy classical channels
which is provably capacity-achieving and has an efficient encoding and
decoding. Recently, this method has been generalized to the realm of quantum
information processing, for tasks such as classical communication, private
classical communication, and quantum communication. In the present work, we
apply the polar coding method to network quantum information theory, by making
use of recent advances for related classical tasks. In particular, we consider
problems such as the compound multiple access channel and the quantum
interference channel. The main result of our work is that it is possible to
achieve the best known inner bounds on the achievable rate regions for these
tasks, without requiring a so-called quantum simultaneous decoder. Thus, our
work paves the way for developing network quantum information theory further
without requiring a quantum simultaneous decoder.Comment: 18 pages, 2 figures, v2: 10 pages, double column, version accepted
for publicatio
Polar Coding for the General Wiretap Channel
Information-theoretic work for wiretap channels is mostly based on random
coding schemes. Designing practical coding schemes to achieve
information-theoretic security is an important problem. By applying the two
recently developed techniques for polar codes, we propose a polar coding scheme
to achieve the secrecy capacity of the general wiretap channel.Comment: Submitted to IEEE ITW 201
Achieving Marton's Region for Broadcast Channels Using Polar Codes
This paper presents polar coding schemes for the 2-user discrete memoryless
broadcast channel (DM-BC) which achieve Marton's region with both common and
private messages. This is the best achievable rate region known to date, and it
is tight for all classes of 2-user DM-BCs whose capacity regions are known. To
accomplish this task, we first construct polar codes for both the superposition
as well as the binning strategy. By combining these two schemes, we obtain
Marton's region with private messages only. Finally, we show how to handle the
case of common information. The proposed coding schemes possess the usual
advantages of polar codes, i.e., they have low encoding and decoding complexity
and a super-polynomial decay rate of the error probability.
We follow the lead of Goela, Abbe, and Gastpar, who recently introduced polar
codes emulating the superposition and binning schemes. In order to align the
polar indices, for both schemes, their solution involves some degradedness
constraints that are assumed to hold between the auxiliary random variables and
the channel outputs. To remove these constraints, we consider the transmission
of blocks and employ a chaining construction that guarantees the proper
alignment of the polarized indices. The techniques described in this work are
quite general, and they can be adopted to many other multi-terminal scenarios
whenever there polar indices need to be aligned.Comment: 26 pages, 11 figures, accepted to IEEE Trans. Inform. Theory and
presented in part at ISIT'1
General Strong Polarization
Arikan's exciting discovery of polar codes has provided an altogether new way
to efficiently achieve Shannon capacity. Given a (constant-sized) invertible
matrix , a family of polar codes can be associated with this matrix and its
ability to approach capacity follows from the {\em polarization} of an
associated -bounded martingale, namely its convergence in the limit to
either or . Arikan showed polarization of the martingale associated with
the matrix to get
capacity achieving codes. His analysis was later extended to all matrices
that satisfy an obvious necessary condition for polarization.
While Arikan's theorem does not guarantee that the codes achieve capacity at
small blocklengths, it turns out that a "strong" analysis of the polarization
of the underlying martingale would lead to such constructions. Indeed for the
martingale associated with such a strong polarization was shown in two
independent works ([Guruswami and Xia, IEEE IT '15] and [Hassani et al., IEEE
IT '14]), resolving a major theoretical challenge of the efficient attainment
of Shannon capacity.
In this work we extend the result above to cover martingales associated with
all matrices that satisfy the necessary condition for (weak) polarization. In
addition to being vastly more general, our proofs of strong polarization are
also simpler and modular. Specifically, our result shows strong polarization
over all prime fields and leads to efficient capacity-achieving codes for
arbitrary symmetric memoryless channels. We show how to use our analyses to
achieve exponentially small error probabilities at lengths inverse polynomial
in the gap to capacity. Indeed we show that we can essentially match any error
probability with lengths that are only inverse polynomial in the gap to
capacity.Comment: 73 pages, 2 figures. The final version appeared in JACM. This paper
combines results presented in preliminary form at STOC 2018 and RANDOM 201
An Entropy Sumset Inequality and Polynomially Fast Convergence to Shannon Capacity Over All Alphabets
We prove a lower estimate on the increase in entropy when two copies of a conditional random variable X | Y, with X supported on Z_q={0,1,...,q-1} for prime q, are summed modulo q. Specifically, given two i.i.d. copies (X_1,Y_1) and (X_2,Y_2) of a pair of random variables (X,Y), with X taking values in Z_q, we show
H(X_1 + X_2 mid Y_1, Y_2) - H(X|Y) >=e alpha(q) * H(X|Y) (1-H(X|Y))
for some alpha(q) > 0, where H(.) is the normalized (by factor log_2(q)) entropy. In particular, if X | Y is not close to being fully random or fully deterministic and H(X| Y) in (gamma,1-gamma), then the entropy of the sum increases by Omega_q(gamma). Our motivation is an effective analysis of the finite-length behavior of polar codes, for which the linear dependence on gamma is quantitatively important. The assumption of q being prime is necessary: for X supported uniformly on a proper subgroup of Z_q we have H(X+X)=H(X). For X supported on infinite groups without a finite subgroup (the torsion-free case) and no conditioning, a sumset inequality for the absolute increase in (unnormalized) entropy was shown by Tao in [Tao, CP&R 2010].
We use our sumset inequality to analyze Ari kan\u27s construction of polar codes and prove that for any q-ary source X, where q is any fixed prime, and anyepsilon > 0, polar codes allow efficient data compression of N i.i.d. copies of X into (H(X)+epsilon)N q-ary symbols, as soon as N is polynomially large in 1/epsilon. We can get capacity-achieving source codes with similar guarantees for composite alphabets, by factoring q into primes and combining different polar codes for each prime in factorization.
A consequence of our result for noisy channel coding is that for all discrete memoryless channels, there are explicit codes enabling reliable communication within epsilon > 0 of the symmetric Shannon capacity for a block length and decoding complexity bounded by a polynomial in 1/epsilon. The result was previously shown for the special case of binary-input channels [Guruswami/Xial, FOCS\u2713; Hassani/Alishahi/Urbanke, CoRR 2013], and this work extends the result to channels over any alphabet
Strong Secrecy on a Class of Degraded Broadcast Channels Using Polar Codes
Different polar coding schemes are proposed for the memoryless degraded
broadcast channel under different reliability and secrecy requirements: layered
decoding and/or layered secrecy. In this setting, the transmitter wishes to
send multiple messages to a set of legitimate receivers keeping them masked
from a set of eavesdroppers. The layered decoding structure requires receivers
with better channel quality to reliably decode more messages, while the layered
secrecy structure requires eavesdroppers with worse channel quality to be kept
ignorant of more messages. The implementation of the proposed polar coding
schemes is discussed and their performance is evaluated by simulations for the
symmetric degraded broadcast channel.Comment: 35 pages. Published in "MDPI Entropy". A short version of this paper
had been accepted to the 3rd Workshop on Physical-Layer Methods for Wireless
Security, IEEE CNS 201
Strong secrecy on a class of degraded broadcast channels using polar codes
Asymptotic secrecy-capacity achieving polar coding schemes are proposed for the memoryless degraded broadcast channel under different reliability and secrecy requirements: layered decoding or layered secrecy. In these settings, the transmitter wishes to send multiple messages to a set of legitimate receivers keeping them masked from a set of eavesdroppers. The layered decoding structure requires receivers with better channel quality to reliably decode more messages, while the layered secrecy structure requires eavesdroppers with worse channel quality to be kept ignorant of more messages. Practical constructions for the proposed polar coding schemes are discussed and their performance evaluated by means of simulations.Peer ReviewedPostprint (published version