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IEICE Transactions on Communications

REP Node Modification and Fast Decoding of Multi-Kernel Polar Codes Constructed with a New 5 × 5 Kernel

Changwei Tu, Youyan Zhang, Zhiliang Huang

Published Apr 17, 2025• 1 citation
Publisher

Abstract

Multi-kernel polar codes, which are constructed by different sizes of kernels make the length of the traditional polar code more flexible. In this paper, we extend the multi-kernel polar codes to 5×5 kernels including kernel design, code design, and fast decoding. First, we construct a new 5×5 kernel with the maximum polarization rate. The first row of this kernel is an all-one vector, which enable us to find the relation between node modification and distance property of multi-kernel polar codes. Second, we modify the repetition (REP) nodes to the new 5×5 kernel, which can increase the distance property of the constructed polar codes, improve the decoding performance and reduce memory requirements during the fast decoding. In addition, we find our REP node modification can be generalized to multi-kernel polar codes when the first row of the used kernels is all-one vector. Third, we prove the four kinds of basic nodes (Rate-1 node, Rate-0 nodes, Repetition (REP) nodes, and Single-Parity check (SPC) nodes) of fast decoding are suitable for the new kernel. Moreover, we generalize the Generalized Parity-Check (G-PC) nodes and the Generalized Repetition (GREP) nodes, which are suitable for multi-kernel polar codes constructed by any size of kernels and any order of kernels. By simulations, we show that our modification improves the error-rate performance of the fast decoding. Meanwhile, the fast decoding of our multi-kernel polar codes has at least 68.4% latency reduction compared with the traditional SC decoder in all cases considered where code lengths are 80, 135, 400, and 675.

DOI
10.23919/transcom.2024ebp3185

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•Jan 1, 2017•0 citations
Multi-kernel polar codes, which are constructed by different sizes of kernels make the length of the traditional polar code more flexible. In this paper, we extend the multi-kernel polar codes to 5×5 kernels including kernel design, code design, and fast decoding. First, we construct a new 5×5 kernel with the maximum polarization rate. The first row of this kernel is an all-one vector, which enable us to find the relation between node modification and distance property of multi-kernel polar codes. Second, we modify the repetition (REP) nodes to the new 5×5 kernel, which can increase the distance property of the constructed polar codes, improve the decoding performance and reduce memory requirements during the fast decoding. In addition, we find our REP node modification can be generalized to multi-kernel polar codes when the first row of the used kernels is all-one vector. Third, we prove the four kinds of basic nodes (Rate-1 node, Rate-0 nodes, Repetition (REP) nodes, and Single-Parity check (SPC) nodes) of fast decoding are suitable for the new kernel. Moreover, we generalize the Generalized Parity-Check (G-PC) nodes and the Generalized Repetition (GREP) nodes, which are suitable for multi-kernel polar codes constructed by any size of kernels and any order of kernels. By simulations, we show that our modification improves the error-rate performance of the fast decoding. Meanwhile, the fast decoding of our multi-kernel polar codes has at least 68.4% latency reduction compared with the traditional SC decoder in all cases considered where code lengths are 80, 135, 400, and 675.
Multi-kernel polar codes, which are constructed by different sizes of kernels make the traditional polar codes have more flexible code length. In this paper, we extend the multi-kernel polar codes to 5×55 \times 55×5 kernels including kernel design, code design, and fast decoding. First, we construct a new 5×55 \times 55×5 kernel, which has the maximum polarization rate. The first row of this kernel is all-one vector, which enable us to seek the relation between node modification and distance property of multi-kernel polar codes. Second, we modify the repetition (REP) nodes to this kernel, which can increase the distance property of the constructed polar codes, improve the decoding performance and reduce memory requirements during the fast decoding. In addition, we find this REP node modification can be generalized when multi-kernel polar codes constructed by kernels, which the first row is all-one vector. Third, we prove the four kinds of basic nodes (Rate-1 node, Rate-0 nodes, Repetition (REP) nodes, and Single-Parity check (SPC) nodes) of fast decoding are suitable for this kernel. Moreover, we generalize the Generalized Parity-Check (G-PC) nodes and the Generalized Repetition (GREP) nodes, which are suitable for multi-kernel polar codes constructed by any size of kernels and any order of kernels. The G-PC and G-REP nodes can further accelerate the decoding speed. By simulations, we show that our modification improves the error-rate performance of the fast decoding. Meanwhile, the fast decoding of our multi-kernel polar codes has at least 68. 4% reduced latency compared with the traditional SC decoder in all cases considered where frame lengths are (80,135,400,67580,135,400,67580,135,400,675). Index Terms-Multi-kernel polar codes, node modification, fast decoding, 5×55 \times 55×5 kernel matrix
Using large size kernels in conjunction with the binary kernel enables length flexibility in polar codes. However, the resultant multi-kernel polar codes suffer from high decoding complexity. The size-3 kernels have drawn much attention due to their sufficiently high polarization exponents and the lowest decoding complexity among larger than size-2 kernels. However, the two specific size-3 kernels that can achieve the optimal polarization have a zero in their last rows. This results in the error-rate performance degradation of the repetition (REP) and generalized-repetition (G-REP) nodes in addition to increased memory requirements for their fast decoders. In this letter, we propose modifications to the REP and G-REP nodes that simultaneously result in improved error-rate performance and reduced memory requirements for their fast decoders.
Polar codes have gained significant a ttention in recent years due to their superior performance, making them an attractive choice for use in modern communication systems. In this paper, we present a novel approach for implementing a fast decoding method for polar codes with additional nodes. Specifically, we propose a modification of the fast decoding method with bit-flipping capability, called the GSCLAF, which is designed to improve the correction capability of the decoder. To achieve this, we consider additional nodes of five types, which are obtained by a combination of basic generalized nodes, namely information, frozen, SPC, and REP nodes. To evaluate the performance of our proposed algorithm, numerical experiments were conducted using the 5G NR polar code. The results of these experiments demonstrate the effectiveness of GSCLAF relative to other popular polar decoding algorithms. Overall, our proposed algorithm provides a powerful tool for improving the decoding performance of polar codes with additional nodes, making it a promising solution for modern communication systems that require high data rates and reliable communication.
The decoding latency of the successive-cancellation decoder for polar codes is of great importance in its practical applications, and it can be reduced by implementing multi-bit decision decoders in the constituent nodes of the decoding tree. In this letter, we develop a matrix analysis method to analyze the corresponding parity submatrix of the decoding tree’s internal node. Two generalized constituent nodes are presented for the fast simplified successive-cancellation (fast-SSC) decoding of linear kernels-mixed multi-kernel polar codes. Results show that the proposed method achieves latency reduction and preserves the error-correction performance.
Non-binary polar codes (NBPCs) decoded by successive cancellation (SC) algorithm have remarkable bit-error-rate performance compared to the binary polar codes (BPCs). Due to its serial nature, SC decoding suffers from long latency. The latency issue in BPCs has been the topic of extensive research and it has been notably resolved by the introduction of fast SC-based decoders. However, the latency problem of NBPCs is mainly untouched and the vast majority of research on NBPCs is devoted to issues concerning design and efficient implementation. In this paper, we propose fast SC decoding for NBPCs constructed based on2×22\times 22×2kernels. In particular, we extend the special nodes of BPCs to their non-binary counterpart and define various non-binary special nodes in the SC decoding tree of NBPCs and propose their fast decoding. This way, we avoid traversing the full decoding tree and significantly reduce the decoding delay compared to symbol-by-symbol SC decoding. We also propose a simplified NBPC structure that facilitates the procedure of non-binary fast SC decoding. Using our proposed fast non-binary decoder, we observe an improvement of more than 96% in latency concerning the original SC decoding. This is while our proposed fast SC decoder for NBPCs incurs no error-rate loss.
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