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IEEE Journal on Selected Areas in Communications

Mixed-Kernels Constructions of Polar Codes

Noam Presman, Ofer Shapira, Simon N. Litsyn

Published Nov 26, 2015• 44 citations
Publisher

Abstract

Mixed kernels of polar codes are mapping functions having inputs of different alphabet sizes that are used to construct polar coding scheme. These schemes are constructed by incorporating several (homogeneous) kernels, each one over different alphabet size. In this paper, the idea of mixed-kernels construction is introduced and analyzed. An asymptotic analysis of the proposed scheme shows that its polarization properties are strongly related to the ones of the constituent kernels. Simulation of finite length instances of the scheme indicate their advantages both in error correction performance and complexity compared to the known polar coding structures.

DOI
10.1109/jsac.2015.2504278

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Mixed kernels of polar codes are mapping functions having inputs of different alphabet sizes that are used to construct polar coding scheme. These schemes are constructed by incorporating several (homogeneous) kernels, each one over different alphabet size. In this paper, the idea of mixed-kernels construction is introduced and analyzed. An asymptotic analysis of the proposed scheme shows that its polarization properties are strongly related to the ones of the constituent kernels. Simulation of finite length instances of the scheme indicate their advantages both in error correction performance and complexity compared to the known polar coding structures.
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.
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.
Polar codes are the first class of error correction codes that achieve the Shannon channel capacity when their code length approaches infinity, but the code lengths of polar codes are limited to powers of 2. Therefore, multi-kernel polar codes are proposed, and their generator matrices are constructed from a variety of polarized kernels with different sizes. In this paper, we study the effect of different kernel orders on the channel polarization in the Gaussian approximation construction for multi-kernel polar codes. Besides, we study the minimum-distance spectrum of polarized kernels and the generator matrices of multi-kernel polar codes, and we calculate the minimum-distance spectrum for some larger kernels. We experimentally compare the advantages and disadvantages of the Gaussian approximation and minimum-distance construction. Finally, we propose a hybrid construction method which considers both the reliability and distance, and experimental results show that the proposed construction method outperforms the construction method that only considers distance and reliability.