1,156 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
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
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
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