803 research outputs found
On the Construction and Decoding of Concatenated Polar Codes
A scheme for concatenating the recently invented polar codes with interleaved
block codes is considered. By concatenating binary polar codes with interleaved
Reed-Solomon codes, we prove that the proposed concatenation scheme captures
the capacity-achieving property of polar codes, while having a significantly
better error-decay rate. We show that for any , and total frame
length , the parameters of the scheme can be set such that the frame error
probability is less than , while the scheme is still
capacity achieving. This improves upon 2^{-N^{0.5-\eps}}, the frame error
probability of Arikan's polar codes. We also propose decoding algorithms for
concatenated polar codes, which significantly improve the error-rate
performance at finite block lengths while preserving the low decoding
complexity
Source and Channel Polarization over Finite Fields and Reed-Solomon Matrices
Polarization phenomenon over any finite field with size
being a power of a prime is considered. This problem is a generalization of the
original proposal of channel polarization by Arikan for the binary field, as
well as its extension to a prime field by Sasoglu, Telatar, and Arikan. In this
paper, a necessary and sufficient condition of a matrix over a finite field
is shown under which any source and channel are polarized.
Furthermore, the result of the speed of polarization for the binary alphabet
obtained by Arikan and Telatar is generalized to arbitrary finite field. It is
also shown that the asymptotic error probability of polar codes is improved by
using the Reed-Solomon matrix, which can be regarded as a natural
generalization of the binary matrix used in the original proposal
by Arikan.Comment: 17 pages, 3 figures, accepted for publication in the IEEE
Transactions on Information Theor
Applications of Algebraic Geometric Codes to Polar Coding
In recent groundbreaking work, Arikan developed polar codes as an explicit construction of symmetric capacity achieving codes for binary discrete memoryless channels with low encoding and decoding complexities. In this construction, a specific kernel matrix G is considered and is used to encode a block of channels. As the number of channels grows, each channel becomes either a noiseless channel or a pure-noise channel, and the rate of this polarization is related to the kernel matrix used. Since Arikan\u27s original construction, polar codes have been generalized to q-ary discrete memoryless channels, where q is a power of a prime, and other matrices have been considered as kernels. In our work, we expand on the ideas of Mori and Tanaka and Korada, Sasoglu, and Urbanke by employing algebraic geometric codes to produce kernels of polar codes, specifically codes from maximal and optimal function fields
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
A characterization of MDS codes that have an error correcting pair
Error-correcting pairs were introduced in 1988 by R. Pellikaan, and were
found independently by R. K\"otter (1992), as a general algebraic method of
decoding linear codes. These pairs exist for several classes of codes. However
little or no study has been made for characterizing those codes. This article
is an attempt to fill the vacuum left by the literature concerning this
subject. Since every linear code is contained in an MDS code of the same
minimum distance over some finite field extension we have focused our study on
the class of MDS codes.
Our main result states that an MDS code of minimum distance has a
-ECP if and only if it is a generalized Reed-Solomon code. A second proof is
given using recent results Mirandola and Z\'emor (2015) on the Schur product of
codes
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