- PII
- 10.31857/S0555292323010011-1
- DOI
- 10.31857/S0555292323010011
- Publication type
- Article
- Status
- Published
- Authors
- Volume/ Edition
- Volume 59 / Issue number 1
- Pages
- 3-16
- Abstract
- Bhattacharyya parameters are used in the theory of polar codes to determine positions of frozen and information bits. These parameters characterize rate of polarization of channels WN(i), 1 ≤ i ≤ N, which are constructed in a special way from the original channel W, where N = 2n is the channel length, n = 1, 2, .... In the case where W is a binary symmetric memoryless channel, we present two series of formulas for the parameters Z(WN(i)): for i = N - 2k + 1, 0 ≤ k ≤ n, and for i = N/2 - 2k + 1, 1 ≤ k ≤ n - 2. The formulas require of the order of $\binom{2^{n-k}+2^k-1}{2^k} 2^{2^k}$ addition operations for the first series and of the order of $\binom{2^{n-k-1}+2^k-1}{2^k} 2^{2^k}$ for the second. In the cases i = 1, N/4 + 1, N/2 + 1, N, the obtained expressions for the parameters have been simplified by computing the sums in them. We show potential generalizations for the values of i in the interval (N/4, N). We also study combinatorial properties of the polarizing matrix GN of a polar code with Arıkan’s kernel. In particular, we establish simple recurrence relations between rows of the matrices GN and GN/2.
- Keywords
- полярный код параметр Бхаттачарьи поляризационная матрица
- Date of publication
- 18.09.2025
- Year of publication
- 2025
- Number of purchasers
- 0
- Views
- 16
References
- 1. Arıkan E. Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels // IEEE Trans. Inform. Theory. 2009. V. 55. № 7. P. 3051-3073. https://doi.org/10.1109/TIT.2009.2021379
- 2. Tal I., Vardy A. How to Construct Polar Codes // IEEE Trans. Inform. Theory. 2013. V. 59. № 10. P. 6542-6582. https://doi.org/10.1109/TIT.2013.2272694
- 3. Sarkis G., Tal I., Giard P., Vardy A., Thibeault C., Gross W.J. Flexible and Low-Complexity Encoding and Decoding of Systematic Polar Codes // IEEE Trans.Commun. 2016. V. 64. № 7. P. 2732-2745. https://doi.org/10.1109/TCOMM.2016.2574996
- 4. Егорычев Г.П. Интегральное представление и вычисление комбинаторных сумм. Новосибирск: Наука, 1977
