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arXiv:2010.07884 · IEEE Transactions on Communications

Window Processing of Binary Polarization Kernels

Grigorii Trofimiuk, Peter Trifonov

Published Apr 12, 2021• 21 citations
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Abstract

A decoding algorithm for polar (sub)codes with binary 2t×2tpolarization kernels is presented. It is based on the window processing (WP) method, which exploits the linear relationship of the polarization kernels and the Arikan matrix. This relationship enables one to compute the kernel input symbols probabilities by computing the probabilities of several paths in Arikan successive cancellation (SC) decoder. In this paper we propose an improved version of WP, which has significantly lower arithmetic complexity and operates in log-likelihood ratios (LLRs) domain. The algorithm identifies and reuses common subexpressions arising in computation of Arikan SC path scores. The proposed algorithm is applied to kernels of size 16 and 32 with improved polarization properties. It enables polar (sub)codes with the considered kernels to simultaneously provide better performance and lower decoding complexity compared with polar (sub)codes with Arikan kernel.

DOI
10.1109/tcomm.2021.3072730

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A decoding algorithm for polar (sub)codes with binary 2t×2tpolarization kernels is presented. It is based on the window processing (WP) method, which exploits the linear relationship of the polarization kernels and the Arikan matrix. This relationship enables one to compute the kernel input symbols probabilities by computing the probabilities of several paths in Arikan successive cancellation (SC) decoder. In this paper we propose an improved version of WP, which has significantly lower arithmetic complexity and operates in log-likelihood ratios (LLRs) domain. The algorithm identifies and reuses common subexpressions arising in computation of Arikan SC path scores. The proposed algorithm is applied to kernels of size 16 and 32 with improved polarization properties. It enables polar (sub)codes with the considered kernels to simultaneously provide better performance and lower decoding complexity compared with polar (sub)codes with Arikan kernel.
A reduced complexity algorithm is presented for computing the log-likelihood ratios (LLRs) arising in the successive cancellation (SC) decoder for polar codes with large kernels of arbitrary dimension. The proposed algorithm exploits a recursive trellis representation of the codes generated by submatrices of the polarization kernel and enables polar codes based on large kernels to be decoded with lower complexity compared to the codes based on the Arıkan kernel with the same performance. Extensions to the case of non-binary kernels are presented.
A method for efficiently successive cancellation (SC) decoding of polar codes with high-dimensional linear binary kernels (HDLBK) is presented and analyzed. We devise a lll-expressions method which can obtain simplified recursive formulas of SC decoder in likelihood ratio form for arbitrary linear binary kernels to reduce the complexity of corresponding SC decoder. By considering the bit-channel transition probabilities WG(⋅)(⋅∣0)W_{G}^{(\cdot)}(\cdot|0)WG(⋅)​(⋅∣0) and WG(⋅)(⋅∣1)W_{G}^{(\cdot)}(\cdot|1)WG(⋅)​(⋅∣1) separately, a WWW-expressions method is proposed to further reduce the complexity of HDLBK based SC decoder. For a m×mm\times mm×m binary kernel, the complexity of straightforward SC decoder is O(2mNlog⁡N)O(2^{m}N\log N)O(2mNlogN). With WWW-expressions, we reduce the complexity of straightforward SC decoder to O(m2Nlog⁡N)O(m^{2}N\log N)O(m2NlogN) when m≤16m\leq 16m≤16. Simulation results show that 16×1616\times1616×16 kernel polar codes offer significant advantages in terms of error performances compared with 2×22\times22×2 kernel polar codes under SC and list SC decoders.
Polar codes constructed by large kernels can attain better finite length performance than those originating from Arıkan’s 2 × 2 kernel. However, the successive cancellation (SC) decoding for these polar codes is impractical even for relatively small kernel size ofmbecause complexity of thekernel computationgrows exponentially withm. This research shows whenm> 2, there exists a large amount oflike termsin the kernel computation which yields a ground for facilitating the decoding. By transferring the kernel computation from the probability domain to the likelihood ratio domain (l-domain), the so-calledl-formula method provides an efficient way to combine the like terms in the kernel computation for kernels up to size 11. However, thel-formula method becomes intractable for kernel size beyond 11. To further reduce the computational complexity, this paper proposes aW-formula method which transforms the kernel computation into the probability pair domain (W-domain). Advanced from thel-domain, the numerator and denominator of the likelihood ratio are considered separately, which eases the restrictions of combining like terms. TheW-formula method can combine much more like terms resulting in a significant reduction on the number of sub-formulas for medium kernels (m≤ 16). Furthermore, in theW-domain, sub-formulas become regular and there exist many common sub-formulas whose computations can be shared. Being able to handle kernels of size up to 16, we show that theW-formula based SC decoding achieves a significant complexity reduction over the existing non-approximate SC decoding (thel-formula based SC decoding).