790 research outputs found
A Packing Lemma for Polar Codes
A packing lemma is proved using a setting where the channel is a binary-input
discrete memoryless channel , the code is
selected at random subject to parity-check constraints, and the decoder is a
joint typicality decoder. The ensemble is characterized by (i) a pair of fixed
parameters where is a parity-check matrix and is a channel
input distribution and (ii) a random parameter representing the desired
parity values. For a code of length , the constraint is sampled from where is the
indicator function of event and . Given , the codewords are chosen conditionally
independently from . It is shown
that the probability of error for this ensemble decreases exponentially in
provided the rate is kept bounded away from
with and . In the special case where is the parity-check
matrix of a standard polar code, it is shown that the rate penalty
vanishes as increases. The paper also discusses the
relation between ordinary polar codes and random codes based on polar
parity-check matrices.Comment: 5 pages. To be presented at 2015 IEEE International Symposium on
Information Theory, June 14-19, 2015, Hong Kong. Minor corrections to v
On privacy amplification, lossy compression, and their duality to channel coding
We examine the task of privacy amplification from information-theoretic and
coding-theoretic points of view. In the former, we give a one-shot
characterization of the optimal rate of privacy amplification against classical
adversaries in terms of the optimal type-II error in asymmetric hypothesis
testing. This formulation can be easily computed to give finite-blocklength
bounds and turns out to be equivalent to smooth min-entropy bounds by Renner
and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a
bound in terms of the divergence by Yang, Schaefer, and Poor
[arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy
amplification based on linear codes can be easily repurposed for channel
simulation. Combined with known relations between channel simulation and lossy
source coding, this implies that privacy amplification can be understood as a
basic primitive for both channel simulation and lossy compression. Applied to
symmetric channels or lossy compression settings, our construction leads to
proto- cols of optimal rate in the asymptotic i.i.d. limit. Finally, appealing
to the notion of channel duality recently detailed by us in [IEEE Trans. Info.
Theory 64, 577 (2018)], we show that linear error-correcting codes for
symmetric channels with quantum output can be transformed into linear lossy
source coding schemes for classical variables arising from the dual channel.
This explains a "curious duality" in these problems for the (self-dual) erasure
channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and
partly anticipates recent results on optimal lossy compression by polar and
low-density generator matrix codes.Comment: v3: updated to include equivalence of the converse bound with smooth
entropy formulations. v2: updated to include comparison with the one-shot
bounds of arXiv:1706.03866. v1: 11 pages, 4 figure
Sphere packing bounds in the Grassmann and Stiefel manifolds
Applying the Riemann geometric machinery of volume estimates in terms of
curvature, bounds for the minimal distance of packings/codes in the Grassmann
and Stiefel manifolds will be derived and analyzed. In the context of
space-time block codes this leads to a monotonically increasing minimal
distance lower bound as a function of the block length. This advocates large
block lengths for the code design.Comment: Replaced with final version, 11 page
Asymmetric Lee Distance Codes for DNA-Based Storage
We consider a new family of codes, termed asymmetric Lee distance codes, that
arise in the design and implementation of DNA-based storage systems and systems
with parallel string transmission protocols. The codewords are defined over a
quaternary alphabet, although the results carry over to other alphabet sizes;
furthermore, symbol confusability is dictated by their underlying binary
representation. Our contributions are two-fold. First, we demonstrate that the
new distance represents a linear combination of the Lee and Hamming distance
and derive upper bounds on the size of the codes under this metric based on
linear programming techniques. Second, we propose a number of code
constructions which imply lower bounds
Sphere packing bounds via spherical codes
The sphere packing problem asks for the greatest density of a packing of
congruent balls in Euclidean space. The current best upper bound in all
sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We
revisit their argument and improve their bound by a constant factor using a
simple geometric argument, and we extend the argument to packings in hyperbolic
space, for which it gives an exponential improvement over the previously known
bounds. Additionally, we show that the Cohn-Elkies linear programming bound is
always at least as strong as the Kabatiansky-Levenshtein bound; this result is
analogous to Rodemich's theorem in coding theory. Finally, we develop
hyperbolic linear programming bounds and prove the analogue of Rodemich's
theorem there as well.Comment: 30 pages, 2 figure
Polar Coding for Secret-Key Generation
Practical implementations of secret-key generation are often based on
sequential strategies, which handle reliability and secrecy in two successive
steps, called reconciliation and privacy amplification. In this paper, we
propose an alternative approach based on polar codes that jointly deals with
reliability and secrecy. Specifically, we propose secret-key capacity-achieving
polar coding schemes for the following models: (i) the degraded binary
memoryless source (DBMS) model with rate-unlimited public communication, (ii)
the DBMS model with one-way rate-limited public communication, (iii) the 1-to-m
broadcast model and (iv) the Markov tree model with uniform marginals. For
models (i) and (ii) our coding schemes remain valid for non-degraded sources,
although they may not achieve the secret-key capacity. For models (i), (ii) and
(iii), our schemes rely on pre-shared secret seed of negligible rate; however,
we provide special cases of these models for which no seed is required.
Finally, we show an application of our results to secrecy and privacy for
biometric systems. We thus provide the first examples of low-complexity
secret-key capacity-achieving schemes that are able to handle vector
quantization for model (ii), or multiterminal communication for models (iii)
and (iv).Comment: 26 pages, 9 figures, accepted to IEEE Transactions on Information
Theory; parts of the results were presented at the 2013 IEEE Information
Theory Worksho
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