# Binary Linear Codes with Optimal Scaling: Polar Codes with Large Kernels

@article{Fazeli2018BinaryLC, title={Binary Linear Codes with Optimal Scaling: Polar Codes with Large Kernels}, author={Arman Fazeli and S. Hamed Hassani and Marco Mondelli and Alexander Vardy}, journal={2018 IEEE Information Theory Workshop (ITW)}, year={2018}, pages={1-5} }

We prove that, at least for the binary erasure channel, the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but, in fact, do so under the best possible scaling of their block length as a function of the gap to capacity. This result exhibits the first known family of binary codes that attain both optimal scaling and quasi-linear complexity of encoding and decoding. Specifically, for any fixed $\delta \gt 0$, we exhibit binary linear codes that ensure reliable…

## 32 Citations

### Explicit Polar Codes with Small Scaling Exponent

- Computer Science2019 IEEE International Symposium on Information Theory (ISIT)
- 2019

A sequence of binary linear codes that approaches capacity on the BEC with quasi-linear complexity and scaling exponent µ < 3.122 is exhibited, which was not previously known to exist.

### Polar Coded Computing: The Role of the Scaling Exponent

- Computer Science2022 IEEE International Symposium on Information Theory (ISIT)
- 2022

This paper reveals a connection between the average execution time and the scaling exponent μ of the family of codes of polar codes and conjecture that these bounds could be improved to O(n–2/μ) and O( n–1), respectively, and provides a heuristic argument as well as numerical evidence supporting this view.

### Sub-4.7 Scaling Exponent of Polar Codes

- Computer ScienceArXiv
- 2022

Polar code visibly approaches channel capacity in practice and is thereby a constituent code of the 5G standard, however, the per- formance of short-length polar code has rooms for improvement that could hinder its adoption by a wider class of applications.

### Polar List Decoding for Large Polarization Kernels

- Computer Science2021 IEEE Globecom Workshops (GC Wkshps)
- 2021

This paper presents a new method that decodes large kernel polar codes with a complexity coefficient that is polynomial to the kernel sizes, and allows us to decode polar codes constructed with a 64 × 64 polarization kernel with scaling exponent µ ≈ 2.87 for the first time.

### Hardness of Successive-Cancellation Decoding of Linear Codes

- Computer Science2020 IEEE International Symposium on Information Theory (ISIT)
- 2020

It is proved that successive-cancellation decoding of general binary linear codes is NP-hard, and every binary linear code can be encoded as a polar code with dynamically frozen bits.

### Accelerating Polarization via Alphabet Extension

- Computer ScienceAPPROX/RANDOM
- 2022

The main contribution is showing that the dynamic of TECs converges to an almost–one-parameter family of channels, which then leads to an upper bound of 3 .

### Polar Codes’ Simplicity, Random Codes’ Durability

- Computer ScienceIEEE Transactions on Information Theory
- 2021

The core theme is to incorporate polar coding with large, random, dynamic kernels (which boosts the performance to random’s realm), and the putative codes are optimal in the following manner.

### List Decoding of Arıkan’s PAC Codes

- Computer Science2020 IEEE International Symposium on Information Theory (ISIT)
- 2020

The main goals in this paper are to answer the following question: is sequential decoding essential for the superior performance of PAC codes and to suggest that the goal of rate-profiling may be to optimize the weight distribution at low weights.

### Polar Coding for Non-Stationary Channels

- Computer ScienceIEEE Transactions on Information Theory
- 2020

The problem of polar coding for an arbitrary sequence of independent binary-input memoryless symmetric (BMS) channels is considered and a polar coding scheme using Arıkan’s channel polarization transform is constructed.

### Parallelism versus Latency in Simplified Successive-Cancellation Decoding of Polar Codes

- Computer Science2021 IEEE International Symposium on Information Theory (ISIT)
- 2021

The tightness of the bound on SSC decoding latency and the applicability of the foregoing results is validated through extensive simulations.

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