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 \(W_N^{(i)}\), \(1\le i\le N\), which are constructed in a special way from the original channel \(W\), where \(N=2^n\) is the channel length, \(n=1,2,\ldots\strut\). In the case where \(W\) is a binary symmetric memoryless channel, we present two series of formulas for the parameters \(Z\bigl(W_N^{(i)}\bigr)\): for \(i=N-2^k+1\), \(0\le k\le n\), and for \(i=N/2-2^k+1\), \(1\le k\le n-2\). 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 \(G_N\) of a polar code with Arıkan’s kernel. In particular, we establish simple recurrence relations between rows of the matrices \(G_N\) and \(G_{N/2}\).
References
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ACKNOWLEDGMENTS
The authors are grateful to a reviewer for useful references and remarks, which helped them to considerably simplify the proofs of the statements in Section 2 and improve the text of the paper in general.
Funding
This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation, Agreement no. 075-02-2022-876.
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Translated from Problemy Peredachi Informatsii, 2023, Vol. 59, No. 1, pp. 3–16. https://doi.org/10.31857/S0555292323010011
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Kolesnikov, S.G., Leontiev, V.M. Series of Formulas for Bhattacharyya Parameters in the Theory of Polar Codes. Probl Inf Transm 59, 1–13 (2023). https://doi.org/10.1134/S0032946023010015
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DOI: https://doi.org/10.1134/S0032946023010015