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arXiv:2310.08369 · IEEE Transactions on Communications

Fast Search Method for Large Polarization Kernels

Grigorii Trofimiuk

Published Oct 16, 2023• 3 citations
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Abstract

A novel search method for large polarization kernels is proposed. The algorithm produces a kernel with given partial distances by employing the depth-first search combined with the computation of coset leaders weight tables and sufficient conditions of code non-equivalence. Using the proposed method, we improved all existing lower bounds on the maximum error exponent for kernels of size from 17 to 29. We also obtained kernels which admit low complexity processing by the recently proposed recursive trellis algorithm. Numerical results demonstrate the advantage of polar codes with the obtained kernels compared with shortened polar codes and polar codes with small kernels.

DOI
10.1109/tcomm.2023.3324651

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IEEE Transactions on Communications•Jun 13, 2023•15 citations
A novel search method for large polarization kernels is proposed. The algorithm produces a kernel with given partial distances by employing the depth-first search combined with the computation of coset leaders weight tables and sufficient conditions of code non-equivalence. Using the proposed method, we improved all existing lower bounds on the maximum error exponent for kernels of size from 17 to 29. We also obtained kernels which admit low complexity processing by the recently proposed recursive trellis algorithm. Numerical results demonstrate the advantage of polar codes with the obtained kernels compared with shortened polar codes and polar codes with small kernels.
A shortening method for large polarization kernels is presented, which results in shortened kernels with the highest error exponent if applied to kernels of size up to 32. It uses lower and upper bounds on partial distances for quick elimination of unsuitable shortening patterns.The proposed algorithm is applied to some kernels of sizes 16 and 32 to obtain shortened kernels of sizes from 9 to 31. These kernels are used in mixed-kernel polar codes of various lengths. Numerical results demonstrate the advantage of polar codes with shortened large kernels compared with shortened and punctured Arikan polar codes, and polar codes with small kernels.
Polar codes constructed with large polarization kernels were recently proven to be able to achieve optimal finite-length scaling properties. However, straightforward decoding for large kernel polar codes introduces a complexity coefficient that is exponential to the kernel sizes, which makes such codes generally believed to be impractical. In this paper, we present a new method that decodes large kernel polar codes with a complexity coefficient that is polynomial to the kernel sizes. This could facilitate the implementation of large kernel polar codes for practical use in the future.Successive cancellation decoding for large kernel polar codes requires calculation on the probabilities of its bit channels. Similar to conventional polar codes, those bit channels follow a recursive relation, which make this calculation boil down to computing the probabilities for bit channels of a single polarization kernel. This kernel-level computation can be shown equivalent to soft-output maximum-likelihood (ML) decoding on a single bit of a linear block code. In our proposed method, we first use linear operations to represent the considered linear block code as a polar code with dynamically frozen bits, and then use a modified polar list decoder to get an approximate value on the soft-output of the desired bit. This method is motivated by the observation that at short block lengths, polar list decoding with a large enough list size can well-approximate ML decoding. The proposed low-complexity method allows us to decode polar codes constructed with a 64 × 64 polarization kernel with scaling exponent µ ≈ 2.87 for the first time.
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.