5 research outputs found
Large Kernel Polar Codes with efficient Window Decoding
In this paper, we modify polar codes constructed with some 2^t x 2^t
polarization kernels to reduce the time complexity of the window decoding. This
modification is based on the permutation of the columns of the kernels. This
method is applied to some of the kernels constructed in the literature of size
16 and 32, with different error exponents and scaling exponents such as eNBCH
kernel. It is shown that this method reduces the complexity of the window
decoding significantly without affecting the performance
Explicit Polar Codes with Small Scaling Exponent
Herein, we focus on explicit constructions of binary kernels
with small scaling exponent for . In particular, we exhibit a
sequence of binary linear codes that approaches capacity on the BEC with
quasi-linear complexity and scaling exponent . To the best of our
knowledge, such a sequence of codes was not previously known to exist. The
principal challenges in establishing our results are twofold: how to construct
such kernels and how to evaluate their scaling exponent.
In a single polarization step, an kernel transforms
an underlying BEC into bit-channels . The erasure
probabilities of , known as the polarization behavior of
, determine the resulting scaling exponent . We first
introduce a class of self-dual binary kernels and prove that their polarization
behavior satisfies a strong symmetry property. This reduces the problem of
constructing to that of producing a certain nested chain of only
self-orthogonal codes. We use nested cyclic codes, whose distance is
as high as possible subject to the orthogonality constraint, to construct the
kernels and . In order to evaluate the polarization behavior
of and , two alternative trellis representations (which may be
of independent interest) are proposed. Using the resulting trellises, we show
that and explicitly compute over half of the polarization
behavior coefficients for , at which point the complexity becomes
prohibitive. To complete the computation, we introduce a Monte-Carlo
interpolation method, which produces the estimate . We
augment this estimate with a rigorous proof that .Comment: Add a reference to G. Trofimiuk and P. Trifonov's pape
Efficient decoding of polar codes with some 1616 kernels
A decoding algorithm for polar codes with binary 1616 kernels with
polarization rate 0.51828 and scaling exponents 3.346 and 3.450 is presented.
The proposed approach exploits the relationship of the considered kernels and
the Arikan matrix to significantly reduce the decoding complexity without any
performance loss. Simulation results show that polar (sub)codes with
1616 kernels can outperform polar codes with Arikan kernel, while
having lower decoding complexity.Comment: This is the extended version of the conference paper. Minor typos are
fixed, arithmetical complexity computations are refine
Convolutional Polar Kernels
A family of polarizing kernels is presented together with
polynomial-complexity algorithm for computing scaling exponent. The proposed
convolutional polar kernels are based on convolutional polar codes, also known
as b-MERA codes. For these kernels, a polynomial-complexity algorithm is
proposed to find weight spectrum of unrecoverable erasure patterns, needed for
computing scaling exponent. As a result, we obtain scaling exponent and
polarization rate for convolutional polar kernels of size up to 1024.Comment: 10 pages, 3 figures. Submitted to IEEE TCO
Window Processing of Binary Polarization Kernels
A decoding algorithm for polar (sub)codes with binary
polarization kernels is presented. It is based on the window processing (WP)
method, which exploits the linear relationship of the polarization kernels and
the Arikan matrix. This relationship enables one to compute the kernel input
symbols probabilities by computing the probabilities of several paths in Arikan
successive cancellation (SC) decoder.
In this paper we propose an improved version of WP, which has significantly
lower arithmetic complexity and operates in log-likelihood ratios (LLRs)
domain. The algorithm identifies and reuses common subexpressions arising in
computation of Arikan SC path scores.
The proposed algorithm is applied to kernels of size 16 and 32 with improved
polarization properties. It enables polar (sub)codes with the considered
kernels to simultaneously provide better performance and lower decoding
complexity compared with polar (sub)codes with Arikan kernel.Comment: Final version to appear in IEEE Transactions on Communications. The
source code is available at https://github.com/gtrofimiuk/SCLKernelDecode