601 research outputs found
Reed-Muller codes for random erasures and errors
This paper studies the parameters for which Reed-Muller (RM) codes over
can correct random erasures and random errors with high probability,
and in particular when can they achieve capacity for these two classical
channels. Necessarily, the paper also studies properties of evaluations of
multi-variate polynomials on random sets of inputs.
For erasures, we prove that RM codes achieve capacity both for very high rate
and very low rate regimes. For errors, we prove that RM codes achieve capacity
for very low rate regimes, and for very high rates, we show that they can
uniquely decode at about square root of the number of errors at capacity.
The proofs of these four results are based on different techniques, which we
find interesting in their own right. In particular, we study the following
questions about , the matrix whose rows are truth tables of all
monomials of degree in variables. What is the most (resp. least)
number of random columns in that define a submatrix having full column
rank (resp. full row rank) with high probability? We obtain tight bounds for
very small (resp. very large) degrees , which we use to show that RM codes
achieve capacity for erasures in these regimes.
Our decoding from random errors follows from the following novel reduction.
For every linear code of sufficiently high rate we construct a new code
, also of very high rate, such that for every subset of coordinates, if
can recover from erasures in , then can recover from errors in .
Specializing this to RM codes and using our results for erasures imply our
result on unique decoding of RM codes at high rate.
Finally, two of our capacity achieving results require tight bounds on the
weight distribution of RM codes. We obtain such bounds extending the recent
\cite{KLP} bounds from constant degree to linear degree polynomials
Properties and Construction of Polar Codes
Recently, Ar{\i}kan introduced the method of channel polarization on which
one can construct efficient capacity-achieving codes, called polar codes, for
any binary discrete memoryless channel. In the thesis, we show that decoding
algorithm of polar codes, called successive cancellation decoding, can be
regarded as belief propagation decoding, which has been used for decoding of
low-density parity-check codes, on a tree graph. On the basis of the
observation, we show an efficient construction method of polar codes using
density evolution, which has been used for evaluation of the error probability
of belief propagation decoding on a tree graph. We further show that channel
polarization phenomenon and polar codes can be generalized to non-binary
discrete memoryless channels. Asymptotic performances of non-binary polar
codes, which use non-binary matrices called the Reed-Solomon matrices, are
better than asymptotic performances of the best explicitly known binary polar
code. We also find that the Reed-Solomon matrices are considered to be natural
generalization of the original binary channel polarization introduced by
Ar{\i}kan.Comment: Master thesis. The supervisor is Toshiyuki Tanaka. 24 pages, 3
figure
Channel Polarization on q-ary Discrete Memoryless Channels by Arbitrary Kernels
A method of channel polarization, proposed by Arikan, allows us to construct
efficient capacity-achieving channel codes. In the original work, binary input
discrete memoryless channels are considered. A special case of -ary channel
polarization is considered by Sasoglu, Telatar, and Arikan. In this paper, we
consider more general channel polarization on -ary channels. We further show
explicit constructions using Reed-Solomon codes, on which asymptotically fast
channel polarization is induced.Comment: 5 pages, a final version of a manuscript for ISIT201
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