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arXiv:1709.10371 · IEEE Information Theory Workshop (ITW)

Multi-kernel polar codes: Proof of polarization and error exponents

Meryem Benammar, Valerio Bioglio, Frédéric Gabry, Ingmar Land

Published Nov 1, 2017• 35 citations
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Abstract

In this paper, we investigate a novel family of polar codes based on multi-kernel constructions, proving that this construction actually polarizes. To this end, we derive a new and more general proof of polarization, which gives sufficient conditions for kernels to polarize. Finally, we derive the convergence rate of the multi-kernel construction and relate it to the convergence rate of each of the constituent kernels.

DOI
10.1109/itw.2017.8277949

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In this paper, we investigate a novel family of polar codes based on multi-kernel constructions, proving that this construction actually polarizes. To this end, we derive a new and more general proof of polarization, which gives sufficient conditions for kernels to polarize. Finally, we derive the convergence rate of the multi-kernel construction and relate it to the convergence rate of each of the constituent kernels.
A new family of multi-kernel polar codes are proposed. Instead of using identical matrices in each stage, assigning different constituent matrices to different virtual channels based on their properties will further improve polarization performance and error rate performance. Matrices with the same block-length can be classified based on Bhattacharyya parameters and a measure M (G) is proposed to compare their polarization performance. Construction framework and graph representation of hybrid multi-kernels are explicitly given. Numerical illustrations show that hybrid multi-kernels have better performance than previous multi-kernels.
Multi-kernel polar codes are constructed from the Kronecker product. For ordinary multi-kernels, constituent polarizing matrices of the same stage are identical. However, in a stage, applying different constituent matrices based on channel property can further improve performance. This family of polar codes is defined as hybrid multi-kernel polar codes. Exponents of hybrid multi-kernels are analyzed. The matrix of a hybrid multi-kernel is generated by a Kronecker-like product operation, and so are partial distances. All exponents of hybrid multi-kernels take values between the maximum and the minimum exponents among all constituent matrices. Exponents are related to the error probability of the original channel and exhibit a trend of convergence as block-length increases. The convergence rate is also related to the original error probability.
We propose a generalized construction for binary polar codes based on mixing multiple kernels of different sizes in order to construct polar codes of block lengths that are not only powers of integers. This results in a multi-kernel polar code with very good performance while the encoding complexity remains low and the decoding follows the same general structure as for the original Arikan polar codes. The construction provides numerous practical advantages as more code lengths can be achieved without puncturing or shortening. We observe numerically that the error-rate performance of our construction outperforms
We propose a generalized construction for binary polar codes based on mixing multiple kernels of different sizes in order to construct polar codes of block lengths that are not only powers of integers. This results in a multi kernel polar code with very good performance while the encoding complexity remains low and the decoding follows the same general structure as for the original Arikan polar codes. The construction provides numerous practical advantages as more code lengths can be achieved without puncturing or shortening. We observe numerically that the error-rate performance of our construction outperforms stateof the-art constructions using puncturing methods.
We propose a generalized construction for binary polar codes based on mixing multiple kernels of different sizes in order to construct polar codes of block lengths that are not only powers of integers. This results in a multi-kernel polar code with very good performance while the encoding complexity remains low and the decoding follows the same general structure as for the original Arikan polar codes. The construction provides numerous practical advantages as more code lengths can be achieved without puncturing or shortening. We observe numerically that the error-rate performance of our construction outperforms state-of-the-art constructions using puncturing methods.
In this paper, code decompositions (a.k.a. code nestings) are used to design binary polarization kernels. The proposed kernels are in general nonlinear. They provide a better polarization exponent than the previously known kernels of the same dimensions. In particular, nonlinear kernels of dimensions 14, 15, and 16 are constructed and are shown to have optimal asymptotic error-correction performance. The optimality is proved by showing that the exponents of these kernels achieve a new upper bound that is developed in this paper.