969 research outputs found
Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions
Polar transforms are central operations in the study of polar codes. This
paper examines polar transforms for non-stationary memoryless sources on
possibly infinite source alphabets. This is the first attempt of source
polarization analysis over infinite alphabets. The source alphabet is defined
to be a Polish group, and we handle the Ar{\i}kan-style two-by-two polar
transform based on the group. Defining erasure distributions based on the
normal subgroup structure, we give recursive formulas of the polar transform
for our proposed erasure distributions. As a result, the recursive formulas
lead to concrete examples of multilevel source polarization with countably
infinite levels when the group is locally cyclic. We derive this result via
elementary techniques in lattice theory.Comment: 12 pages, 1 figure, a short version has been accepted by the 2019
IEEE International Symposium on Information Theory (ISIT2019
R\'enyi Bounds on Information Combining
Bounds on information combining are entropic inequalities that determine how
the information, or entropy, of a set of random variables can change when they
are combined in certain prescribed ways. Such bounds play an important role in
information theory, particularly in coding and Shannon theory. The arguably
most elementary kind of information combining is the addition of two binary
random variables, i.e. a CNOT gate, and the resulting quantities are
fundamental when investigating belief propagation and polar coding. In this
work we will generalize the concept to R\'enyi entropies. We give optimal
bounds on the conditional R\'enyi entropy after combination, based on a certain
convexity or concavity property and discuss when this property indeed holds.
Since there is no generally agreed upon definition of the conditional R\'enyi
entropy, we consider four different versions from the literature. Finally, we
discuss the application of these bounds to the polarization of R\'enyi
entropies under polar codes.Comment: 14 pages, accepted for presentation at ISIT 202
Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels
A method is proposed, called channel polarization, to construct code
sequences that achieve the symmetric capacity of any given binary-input
discrete memoryless channel (B-DMC) . The symmetric capacity is the highest
rate achievable subject to using the input letters of the channel with equal
probability. Channel polarization refers to the fact that it is possible to
synthesize, out of independent copies of a given B-DMC , a second set of
binary-input channels such that, as becomes
large, the fraction of indices for which is near 1
approaches and the fraction for which is near 0
approaches . The polarized channels are
well-conditioned for channel coding: one need only send data at rate 1 through
those with capacity near 1 and at rate 0 through the remaining. Codes
constructed on the basis of this idea are called polar codes. The paper proves
that, given any B-DMC with and any target rate , there
exists a sequence of polar codes such that
has block-length , rate , and probability of
block error under successive cancellation decoding bounded as P_{e}(N,R) \le
\bigoh(N^{-\frac14}) independently of the code rate. This performance is
achievable by encoders and decoders with complexity for each.Comment: The version which appears in the IEEE Transactions on Information
Theory, July 200
Polar Coding for Secure Transmission and Key Agreement
Wyner's work on wiretap channels and the recent works on information
theoretic security are based on random codes. Achieving information theoretical
security with practical coding schemes is of definite interest. In this note,
the attempt is to overcome this elusive task by employing the polar coding
technique of Ar{\i}kan. It is shown that polar codes achieve non-trivial
perfect secrecy rates for binary-input degraded wiretap channels while enjoying
their low encoding-decoding complexity. In the special case of symmetric main
and eavesdropper channels, this coding technique achieves the secrecy capacity.
Next, fading erasure wiretap channels are considered and a secret key agreement
scheme is proposed, which requires only the statistical knowledge of the
eavesdropper channel state information (CSI). The enabling factor is the
creation of advantage over Eve, by blindly using the proposed scheme over each
fading block, which is then exploited with privacy amplification techniques to
generate secret keys.Comment: Proceedings of the 21st Annual IEEE International Symposium on
Personal, Indoor, and Mobile Radio Communications (PIMRC 2010), Sept. 2010,
Istanbul, Turke
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