This paper proposes a generalization of the recently introduced successive cancellation flip (SCFlip) decoding of polar codes, characterized by a number of extra decoding attempts, where one or several positions are flipped from the standard SC decoding. To make such an approach effective, we first introduce the concept of higher order bit flips and propose a new metric to determine the bit flips that are more likely to correct the trajectory of the SC decoding. We then propose a generalized SCFlip decoding algorithm, referred to as dynamic-SCFlip (D-SCFlip), which dynamically builds a list of candidate bit flips, while guaranteeing that the next attempt has the highest probability of success among the remaining ones. Simulation results show that D-SCFlip is an effective alternative to SC-list decoding of polar codes, by providing very good error correcting performance, with an average computation complexity close to the one of the SC decoder.

Dynamic-SCFlip Decoding of Polar Codes

Puncturing is widely used to generate rate-compatible codes. However, for punctured polar codes, some puncturing patterns may greatly affect the split bit channels and cause considerable performance loss. In this paper, we aim to investigate how the split bit channels are affected by various puncturing patterns, and then evaluate the performances of these patterns. We propose a search algorithm to design good punctured polar codes, and prove that designing the optimal puncturing pattern for output bits is equivalent to finding the optimal puncturing pattern for frozen bits. We also propose a heuristic approach based on the idea of polarization to select the position of each punctured output bit, and simulations show that the puncturing pattern designed this way almost achieves the same performance as the optimal one selected by exhaustive search.

On the puncturing patterns for punctured polar codes

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.

Multi-kernel construction of polar codes

Three areas of ongoing research in channel coding are surveyed, and recent developments are presented in each area: Spatially coupled low-density parity-check (LDPC) codes, non-binary LDPC codes, and polar coding.

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Challenges and some new directions in channel coding

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.

Mixed-Kernels Constructions of Polar Codes

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.

Binary Polarization Kernels From Code Decompositions

Code decompositions (a.k.a code nestings) are used to design good binary polar code kernels. The proposed kernels are in general non-linear and show a better rate of polarization under successive cancelation decoding, than the ones suggested by Korada et al., for the same kernel dimensions. In particular, we construct kernels of sizes 14, 15 and 16 providing polarization rates better than any binary kernel of such sizes.

Binary polar code kernels from code decompositions

A generalization of the polar coding scheme called mixed-kernels is introduced. This generalization exploits several homogeneous kernels over alphabets of different sizes. 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.

Polar Codes with Mixed-Kernels

A generalization of the polar coding scheme is proposed. It exploits several homogeneous kernels over alphabets of different sizes. An analysis of the introduced scheme is undertaken. Specifically, asymptotic properties of the polarization are shown to be strongly related to the ones of the constituent kernels.

Ofer Shapira

Papers

Mixed-Kernels Constructions of Polar Codes

Binary Polarization Kernels From Code Decompositions

Binary polar code kernels from code decompositions

Polar Codes with Mixed-Kernels

Polar codes with mixed kernels