A series of formulas for Bhattacharya parameters in the theory of polar codes

S. G. Kolesnikov ; V. M. Leontiev

Prikladnaya Diskretnaya Matematika. Supplement, no. 15 (2022), pp. 108-109.

Résumé

In the theory of polar codes, the Bhattacharya parameters are used to determine the positions of frozen and information bits. The parameters characterize the polarization rate of the channels $W_N^{(i)}$ constructed in a special way from the original channel $W$, here $1 \leqslant i \leqslant N$, $N=2^n$, and $n=1,2, \ldots$ is the length of the code. It is assumed that the $i$-th bit of a message is transmitted over the channel $W_N^{(i)}$, and the Bhattacharya parameter $Z(W_N^{(i)})$ can be interpreted as the noise level of $W_N^{(i)}$. $W$ is a model of a physical transmission channel. If $W$ is a classical binary memoryless symmetric channel, the currently known formulas for the Bhattacharya parameters contain $2^N=2^{2^n}$ terms. We have obtained the formulas for the series of channels $W_N^{(N-2^k+1)}$, $k=0,1, \ldots, n-1$, that contain $2^{(n-k+1)2^k}$ terms. Some assumptions are also given for further simplification of the obtained formulas.

Détail

Mots-clés : polar code
Keywords: Bhattacharya parameter.

Bibliographie

[1] Arikan E., Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels, 2009, arXiv: 0807.3917 | MR

[2] Arikan E., “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels”, IEEE Trans. Inform. Theory, 55:7 (2009), 3051–3073 | DOI | MR | Zbl