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Explicit Polar Codes with Small Scaling Exponent. (arXiv:1901.08186v2 [cs.IT] UPDATED)
来源于:arXiv
Herein, we focus on explicit constructions of $\ell\times\ell$ binary kernels
with small scaling exponent for $\ell \le 64$. In particular, we exhibit a
sequence of binary linear codes that approaches capacity on the BEC with
quasi-linear complexity and scaling exponent $\mu < 3$. 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 $\ell\times\ell$ kernel $K_\ell$ transforms
an underlying BEC into $\ell$ bit-channels $W_1,W_2,\ldots,W_\ell$. The erasure
probabilities of $W_1,W_2,\ldots,W_\ell$, known as the polarization behavior of
$K_\ell$, determine the resulting scaling exponent $\mu(K_\ell)$. 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 $K_\ 查看全文>>