3,263 research outputs found
Algebraic Properties of Polar Codes From a New Polynomial Formalism
Polar codes form a very powerful family of codes with a low complexity
decoding algorithm that attain many information theoretic limits in error
correction and source coding. These codes are closely related to Reed-Muller
codes because both can be described with the same algebraic formalism, namely
they are generated by evaluations of monomials. However, finding the right set
of generating monomials for a polar code which optimises the decoding
performances is a hard task and channel dependent. The purpose of this paper is
to reveal some universal properties of these monomials. We will namely prove
that there is a way to define a nontrivial (partial) order on monomials so that
the monomials generating a polar code devised fo a binary-input symmetric
channel always form a decreasing set.
This property turns out to have rather deep consequences on the structure of
the polar code. Indeed, the permutation group of a decreasing monomial code
contains a large group called lower triangular affine group. Furthermore, the
codewords of minimum weight correspond exactly to the orbits of the minimum
weight codewords that are obtained from (evaluations) of monomials of the
generating set. In particular, it gives an efficient way of counting the number
of minimum weight codewords of a decreasing monomial code and henceforth of a
polar code.Comment: 14 pages * A reference to the work of Bernhard Geiger has been added
(arXiv:1506.05231) * Lemma 3 has been changed a little bit in order to prove
that Proposition 7.1 in arXiv:1506.05231 holds for any binary input symmetric
channe
Polar Codes with exponentially small error at finite block length
We show that the entire class of polar codes (up to a natural necessary
condition) converge to capacity at block lengths polynomial in the gap to
capacity, while simultaneously achieving failure probabilities that are
exponentially small in the block length (i.e., decoding fails with probability
for codes of length ). Previously this combination
was known only for one specific family within the class of polar codes, whereas
we establish this whenever the polar code exhibits a condition necessary for
any polarization.
Our results adapt and strengthen a local analysis of polar codes due to the
authors with Nakkiran and Rudra [Proc. STOC 2018]. Their analysis related the
time-local behavior of a martingale to its global convergence, and this allowed
them to prove that the broad class of polar codes converge to capacity at
polynomial block lengths. Their analysis easily adapts to show exponentially
small failure probabilities, provided the associated martingale, the ``Arikan
martingale'', exhibits a corresponding strong local effect. The main
contribution of this work is a much stronger local analysis of the Arikan
martingale. This leads to the general result claimed above.
In addition to our general result, we also show, for the first time, polar
codes that achieve failure probability for any
while converging to capacity at block length polynomial in the gap to capacity.
Finally we also show that the ``local'' approach can be combined with any
analysis of failure probability of an arbitrary polar code to get essentially
the same failure probability while achieving block length polynomial in the gap
to capacity.Comment: 17 pages, Appeared in RANDOM'1
Towards practical minimum-entropy universal decoding
Minimum-entropy decoding is a universal decoding algorithm used in decoding block compression of discrete memoryless sources as well as block transmission of information across discrete memoryless channels. Extensions can also be applied for multiterminal decoding problems, such as the Slepian-Wolf source coding problem. The 'method of types' has been used to show that there exist linear codes for which minimum-entropy decoders achieve the same error exponent as maximum-likelihood decoders. Since minimum-entropy decoding is NP-hard in general, minimum-entropy decoders have existed primarily in the theory literature. We introduce practical approximation algorithms for minimum-entropy decoding. Our approach, which relies on ideas from linear programming, exploits two key observations. First, the 'method of types' shows that that the number of distinct types grows polynomially in n. Second, recent results in the optimization literature have illustrated polytope projection algorithms with complexity that is a function of the number of vertices of the projected polytope. Combining these two ideas, we leverage recent results on linear programming relaxations for error correcting codes to construct polynomial complexity algorithms for this setting. In the binary case, we explicitly demonstrate linear code constructions that admit provably good performance
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
Information Theoretic Security for Broadcasting of Two Encrypted Sources under Side-Channel Attacks
We consider the secure communication problem for broadcasting of two
encrypted sources. The sender wishes to broadcast two secret messages via two
common key cryptosystems. We assume that the adversary can use the
side-channel, where the side information on common keys can be obtained via the
rate constraint noiseless channel. To solve this problem we formulate the post
encryption coding system. On the information leakage on two secrete messages to
the adversary, we provide an explicit sufficient condition to attain the
exponential decay of this quantity for large block lengths of encrypted
sources.Comment: 13 pages, 4 figures. In the current version we we have corrected
errors in Fig. 2 and Fig. 4. arXiv admin note: substantial text overlap with
arXiv:1801.02563, arXiv:1801.0492
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