关于Polar码重量谱的一个验证
A Verification of Polar Code Weight Spectrum
摘要: Polar码的优势和局限性都在于其严格的结构。Polar码常用的信息位选择方式为根据串行抵消(SC)译码算法中的信道可靠度进行选择，但是这样的选择方法不适用所有的情况。在基于SC译码算法的基础上改进的列表(SCL)译码算法中，选择信息位需要考虑码重量谱和可靠度，得到的性能较好。目前，已经有很多通过改善码谱进而改善性能的方法了。为了去寻找一种通用的改善码谱的方法，我们需要刻画原码重量谱的分布规则，探索在不同情形下选择信息位的方法。在本文中，提出了一种新的行表示方法，在此基础上验证了码谱唯一的信息集的数量范围，引入容斥原理推出二元域下码重的通用公式。
Abstract: The advantages and disadvantages of Polar codes lie in its strict structure. At present, the existing information set selection method is based on the reliability of Successive Cancellation (SC) decoding algorithm, which may not be suitable for other algorithms. Improved SCL (SC List) decoding based on SC decoding needs to consider reliability and code weight spectrum. Various methods to improve the spectrum have been proposed and achieved better performance. In order to find a good way to improve the spectrum, we characterize the distribution law of weight spectrum and try to find suitable information set selection methods in different situations. In this paper, a new row representation method is proposed. On this basis, the number range of the information set of the unique code spectrum is verified, and the general formula of code weight in the binary field is derived by introducing the principle of tolerance and exclusion.
文章引用：乔玉洁, 杨卫华. 关于Polar码重量谱的一个验证[J]. 应用数学进展, 2022, 11(3): 956-965. https://doi.org/10.12677/AAM.2022.113102

参考文献

[1] Arikan, E. (2009) Channel Polarization: A Method for Constructing Capacity Achieving Codes for Symmetric Binary-Input Memoryless Channels. IEEE Transactions on Information Theory, 55, 3051-3073. [Google Scholar] [CrossRef] 
[2] Tae, I. and Vardy, A. (2012) List Decoding of Polar Codes. arXiv: 1206.0050v1.
[3] Li, B., Shen, H. and Tse, D. (2014) A RM-Polar Codes. Mathematics. arXiv:1407.5483v1.
[4] Niu, K. and Cien, K. (2012) CRC-Aided Decoding of Polar Codes. IEEE Communications Letters, 16, 1668-1671. [Google Scholar] [CrossRef] 
[5] Zhang, H.Z., Li, R., Wang, J., Dai, S., Zhang, G., Chen, Y., et al. (2018) Parity-Check Polar Coding for 5G and Beyond. 2018 IEEE International Conference on Communications, Kansas City, 20-24 May 2018, 1-7. [Google Scholar] [CrossRef] 
[6] Arikan, E. (1908) From Sequential Decoding to Channel Polarization and Back again. arXiv: 1908.09594v1.
[7] Li, B., Zhang, H. and Gu, J. (2019) On Pre-Transformed Polar Codes. arXiv:1912.06359v1.
[8] Bardet, M., Dragoi, V., Otmani, A. and Tillich, J.-P. (2016) Algebraic Properties of Polar Codes from a New Polynomial Formalism. Proc. of the IEEE Int. Symposium on Inform. Theory (ISIT), Barcelona, 10-15 July 2016, 230-234. [Google Scholar] [CrossRef] 
[9] Bioglio, V. and Land, I. (2018) Polar Codes with Internal Edge Permutations. 2018 IEEE Wireless Communications and Networking Conference Workshops (WCNCW), Barcelona, 15-18 April 2018, 61-66. [Google Scholar] [CrossRef] 
[10] Chiu, M.C. (2019) Interleaved Polar (I-Polar) Codes. IEEE Transactions on Information Theory, 66, 2430-2442. [Google Scholar] [CrossRef] 
[11] Chiu, M.C. and Chen, C.Y. (2020) Design of I-Polar Codes. IEEE Communications Letters, 24, 1865-1869. [Google Scholar] [CrossRef] 
[12] Tal, I. and Vardy, A. (2013) How to Construct Polar Codes. IEEE Transactions on Information Theory, 59, 6562-6582. [Google Scholar] [CrossRef] 
[13] Trifonov, P. (2012) Efficient Design and Decoding of Polar Codes. IEEE Transactions on Communications, 60, 3221-3227. [Google Scholar] [CrossRef] 
[14] Elkelesh, A., Ebada, M., Cammerer, S. and Ten Brink, S. (2019) Decoder Tailored Polar Code Design Using the Genetic Algorithm. IEEE Transactions on Communications, 67, 4521-4534. [Google Scholar] [CrossRef] 
[15] He, G., Belfiore, J. C., Land, I., Yang, G., Liu, X., Chen, Y., Li, R., Wang, J., Ge, Y., Zhang, R. and Tong, W. (2017) Beta-Expansion: A Theoretical Framework for Fast and Recursive Construction of Polar Codes. GLOBECOM 2017: 2017 IEEE Global Communications Conference, Singapore, 4-8 December 2017, 1-6. [Google Scholar] [CrossRef] 
[16] Ebada, M., Cammerer, S., Elkelesh, A. and Ten Brink, S. (2019) Deep Learning Based Polar Code Design. 2019 57th Annual Allerton Conference on Communication, Control, and Computing, Monticello, 24-27 September 2019, 177-183. [Google Scholar] [CrossRef] 

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