54,051 research outputs found
Universal Polar Codes for More Capable and Less Noisy Channels and Sources
We prove two results on the universality of polar codes for source coding and
channel communication. First, we show that for any polar code built for a
source there exists a slightly modified polar code - having the same
rate, the same encoding and decoding complexity and the same error rate - that
is universal for every source when using successive cancellation
decoding, at least when the channel is more capable than
and is such that it maximizes for the given channels
and . This result extends to channel coding for discrete
memoryless channels. Second, we prove that polar codes using successive
cancellation decoding are universal for less noisy discrete memoryless
channels.Comment: 10 pages, 3 figure
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
Universal Polarization
A method to polarize channels universally is introduced. The method is based
on combining two distinct channels in each polarization step, as opposed to
Arikan's original method of combining identical channels. This creates an equal
number of only two types of channels, one of which becomes progressively better
as the other becomes worse. The locations of the good polarized channels are
independent of the underlying channel, guaranteeing universality. Polarizing
the good channels further with Arikan's method results in universal polar codes
of rate 1/2. The method is generalized to construct codes of arbitrary rates.
It is also shown that the less noisy ordering of channels is preserved under
polarization, and thus a good polar code for a given channel will perform well
over a less noisy one.Comment: Submitted to the IEEE Transactions on Information Theor
Construction of Polar Codes with Sublinear Complexity
Consider the problem of constructing a polar code of block length for the
transmission over a given channel . Typically this requires to compute the
reliability of all the synthetic channels and then to include those that
are sufficiently reliable. However, we know from [1], [2] that there is a
partial order among the synthetic channels. Hence, it is natural to ask whether
we can exploit it to reduce the computational burden of the construction
problem.
We show that, if we take advantage of the partial order [1], [2], we can
construct a polar code by computing the reliability of roughly a fraction
of the synthetic channels. In particular, we prove that
is a lower bound on the number of synthetic channels to be
considered and such a bound is tight up to a multiplicative factor . This set of roughly synthetic channels is universal, in
the sense that it allows one to construct polar codes for any , and it can
be identified by solving a maximum matching problem on a bipartite graph.
Our proof technique consists of reducing the construction problem to the
problem of computing the maximum cardinality of an antichain for a suitable
partially ordered set. As such, this method is general and it can be used to
further improve the complexity of the construction problem in case a new
partial order on the synthetic channels of polar codes is discovered.Comment: 9 pages, 3 figures, presented at ISIT'17 and submitted to IEEE Trans.
Inform. Theor
- …