• Corpus ID: 56475951

Polar-like Codes and Asymptotic Tradeoff among Block Length, Code Rate, and Error Probability

@article{Wang2018PolarlikeCA,
  title={Polar-like Codes and Asymptotic Tradeoff among Block Length, Code Rate, and Error Probability},
  author={Hsin-Po Wang and Iwan M. Duursma},
  journal={ArXiv},
  year={2018},
  volume={abs/1812.08112}
}
A general framework is proposed that includes polar codes over arbitrary channels with arbitrary kernels. The asymptotic tradeoff among block length $N$, code rate $R$, and error probability $P$ is analyzed. Given a tradeoff between $N,P$ and a tradeoff between $N,R$, we return an interpolating tradeoff among $N,R,P$ (Theorem 5). $\def\Capacity{\text{Capacity}}$Quantitatively, if $P=\exp(-N^{\beta^*})$ is possible for some $\beta^*$ and if $R=\Capacity-N^{1/\mu^*}$ is possible for some $1/\mu… 

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References

SHOWING 1-10 OF 81 REFERENCES

Binary Linear Codes with Optimal Scaling and Quasi-Linear Complexity

TLDR
This work presents the first family of binary codes that attains optimal scaling and quasi-linear complexity, at least for the binary erasure channel (BEC), and proves that there exist $\ell\times\ell$ binary kernels, such that polar codes constructed from these kernels achieve scaling exponent that tends to the optimal value of $2$ as $\ell$ grows.

Exponents of polar codes using algebraic geometric code kernels

TLDR
This paper employs more general algebraic geometric (AG) codes to produce kernels of polar codes, and demonstrates that both Hermitian and Suzuki kernels have larger exponents than Reed–Solomon codes over the same field, however, the largerexponents are at the expense of larger kernel matrices.

Using concatenated algebraic geometry codes in channel polarization

TLDR
This paper studies the performance of binary polar codes that are obtained from non-binary algebraic geometry codes using concatenation and finds that for each binary kernels of a given length $n$ there is an optimal choice.

Relaxed Polar Codes

Polar codes are the latest breakthrough in coding theory, as they are the first family of codes with explicit construction that provably achieve the symmetric capacity of binary-input discrete

Unified scaling of polar codes: Error exponent, scaling exponent, moderate deviations, and error floors

TLDR
A unified framework to characterize the relationship between R, N, P<sub>e</sub>, and W is developed and it is proved that polar codes are not affected by error floors.

Polar Code Moderate Deviation: Recovering the Scaling Exponent

TLDR
It is shown that polar coding is able to produce a series of codes that achieve capacity on symmetric binary-input memoryless channels and the latest result in the moderate deviation regime does not imply the scaling exponent regime as a special case.

Polar Codes with Exponentially Small Error at Finite Block Length

TLDR
This work shows that the entire class of polar codes converge to capacity at block lengths polynomial in the gap to capacity, while simultaneously achieving failure probabilities that are exponentially small in the block length (i.e., decoding fails with probability $\exp(-N^{\Omega(1)})$ for codes of length $N$).

Scaling Exponent of List Decoders With Applications to Polar Codes

TLDR
It is shown that under MAP decoding, although the introduction of a list can significantly improve the involved constants, the scaling exponent itself, i.e., the speed at which capacity is approached, stays unaffected for any finite list size.

Near-optimal finite-length scaling for polar codes over large alphabets

  • H. PfisterR. Urbanke
  • Computer Science
    2016 IEEE International Symposium on Information Theory (ISIT)
  • 2016
TLDR
The primary result is that, for any γ > 0 and δ > 0, there is a q<sub>0</sub> such that the fraction of effective channels with erasure rate at most N<sup>-γ</sup> is at least 1 - ε - O(N<Sup>-1/2+δ</sup>.

On Finite-Length Performance of Polar Codes: Stopping Sets, Error Floor, and Concatenated Design

TLDR
A polar code-based concatenated scheme to be used in Optical Transport Networks (OTNs) is proposed and it is shown that the proposed scheme outperforms the existing methods by closing the gap to the capacity while avoiding error floor, and maintaining a low complexity at the same time.
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