1,589 research outputs found
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
Concatenated Polar Codes
Polar codes have attracted much recent attention as the first codes with low
computational complexity that provably achieve optimal rate-regions for a large
class of information-theoretic problems. One significant drawback, however, is
that for current constructions the probability of error decays
sub-exponentially in the block-length (more detailed designs improve the
probability of error at the cost of significantly increased computational
complexity \cite{KorUS09}). In this work we show how the the classical idea of
code concatenation -- using "short" polar codes as inner codes and a
"high-rate" Reed-Solomon code as the outer code -- results in substantially
improved performance. In particular, code concatenation with a careful choice
of parameters boosts the rate of decay of the probability of error to almost
exponential in the block-length with essentially no loss in computational
complexity. We demonstrate such performance improvements for three sets of
information-theoretic problems -- a classical point-to-point channel coding
problem, a class of multiple-input multiple output channel coding problems, and
some network source coding problems
High-Girth Matrices and Polarization
The girth of a matrix is the least number of linearly dependent columns, in
contrast to the rank which is the largest number of linearly independent
columns. This paper considers the construction of {\it high-girth} matrices,
whose probabilistic girth is close to its rank. Random matrices can be used to
show the existence of high-girth matrices with constant relative rank, but the
construction is non-explicit. This paper uses a polar-like construction to
obtain a deterministic and efficient construction of high-girth matrices for
arbitrary fields and relative ranks. Applications to coding and sparse recovery
are discussed
- β¦