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A Note on the Mixing Factor of Polar Codes
Keer Wei1, Xiaoyu Jin1, Weihua Yang1
1School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China.
Entropy (Basel, Switzerland)
|November 24, 2023
Summary
Polar codes achieve near maximum likelihood performance with successive cancellation list decoding when the list size is sufficient. This study establishes a new achievable upper bound for the mixing factor in polar codes with even m and proposes an improved decoding rule.
Area of Science:
- Coding Theory
- Information Theory
- Digital Communications
Background:
- Polar codes are a class of error-correcting codes known for their ability to achieve the theoretical channel capacity.
- Successive Cancellation List (SCL) decoding is a powerful decoding algorithm for polar codes, approaching Maximum Likelihood (ML) performance.
- The mixing factor (MF) is a crucial parameter influencing the performance of polar codes, particularly concerning the list size required for ML-like performance.
Purpose of the Study:
- To determine an achievable upper bound for the mixing factor of polar codes with length n=2^m and rate R<=1/2 when m is an even number.
- To propose a novel hard-decision decoding rule for polar codes that extends beyond the last frozen bit.
- To enhance the understanding and practical application of polar codes in binary-input memoryless symmetric (BMS) channels.
Main Methods:
- Analysis of the mixing factor (MF) for polar codes with specific code lengths and rates.
- Derivation of an achievable upper bound for the MF in the case where m is even.
- Development of a new hard-decision decoding rule for SCL decoding of polar codes.
Main Results:
- An achievable upper bound for the mixing factor (MF) of decreasing monomial polar codes with length n=2^m and rate R<=1/2 is established for even m.
- The previously known upper bound for odd m by Yao et al. is confirmed to be reachable, and the new bound for even m is also shown to be achievable.
- A new hard-decision decoding rule is proposed, offering potential improvements in decoding performance beyond the last frozen bit.
Conclusions:
- The study provides a more complete understanding of the mixing factor's upper bounds for polar codes across different parities of m.
- The proposed decoding rule offers a new method for enhancing the performance of SCL decoding, particularly in the later stages of the decoding process.
- These findings contribute to the advancement of polar coding techniques for reliable communication over BMS channels.
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