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arXiv:1307.2352 · IEEE Information Theory Workshop (ITW)

Polar codes with dynamic frozen symbols and their decoding by directed search

Peter Trifonov, Vera Miloslavskaya

Published Sep 1, 2013• 104 citations
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Abstract

A novel construction of polar codes with dynamic frozen symbols is proposed. The proposed codes are subcodes of extended BCH codes, which ensure sufficiently high minimum distance. Furthermore, a decoding algorithm is proposed, which employs estimates of the not-yet-processed bit channel error probabilities to perform directed search in code tree, reducing thus the total number of iterations.

DOI
10.1109/itw.2013.6691213

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A novel construction of polar codes with dynamic frozen symbols is proposed. The proposed codes are subcodes of extended BCH codes, which ensure sufficiently high minimum distance. Furthermore, a decoding algorithm is proposed, which employs estimates of the not-yet-processed bit channel error probabilities to perform directed search in code tree, reducing thus the total number of iterations.
The eBCH-polar codes is a powerful way to construct polar codes. With high minimum Hamming distance, it outperforms ordinary Polar codes under successive cancellation list (SCL) decoder. However, the absence of cyclic redundancy check (CRC) bits in eBCH-polar codes makes it challenging to adopt SCL-flip (SCLF) decoding scheme. In this research, we propose a flip scheme activated by a Normalized Path Metric (PM) range and a mismatch criterion, which respectively leverage the properties of SCL decoding and dynamic frozen bits of eBCH-polar codes. This work also improves the selection the flipping bits by modifying metric to account for the structure of dynamic frozen bits and reduce the search space. Based on these modifications, we proposed SCLF-1 and SCLF-2 decoding algorithms for eBCH-polar codes. Simulation results show that both of the proposed decoding schemes achieve superior performance compared to other contemporary decoding algorithms for CRC aided polar (CA-polar) codes, demonstrating enhanced error rates performance within the same decoding attempt.
In the context of decoding binary linear block codes by polar code decoding techniques, we propose in this paper a new optimization of the serial nature of decoding the polar codes equivalent to binary linear block codes. In addition to the special nodes proposed by the simplified successive-cancellation list technique, we propose a new special node allowing to estimate in parallel the bits of its sub-code. The simulation is done in an additive white gaussian noise channel (AWGN) channel for several linear block codes, namely bose–chaudhuri–hocquenghem codes (BCH) codes, quadratic-residue (QR) codes, and linear block codes recently designed in the literature. The performance of the proposed technique offers the same performance in terms of frame error rate (FER) as the ordered statistics decoding (OSD) algorithm, which achieves that of maximum likelihood decoder (MLD), but with high memory requirements and computational complexity.
A lower bound on the minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix. Moreover, a construction of convolutional polar subcodes is proposed, which provides improved performance under successive cancellation list decoding. For sufficiently large list size, the decoding complexity of convolutional polar subcodes appears to be lower compared with Arikan polar subcodes with the same performance. The error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of Arikan polar subcodes with the same list size.
In the construction of polar code, the selection of frozen bits affects the error-correcting performance significantly. Several calculation-based algorithms have been proposed for general binary-input discrete memoryless channels (B-DMCs) like the additive white Gaussian noise (AWGN) channel. In this paper, a method for frozen bits selection based on Monte Carlo simulation and belief propagation (BP) decoding is proposed. The information bits are selected out one by one incrementally. The numerical results show that the proposed algorithm can effectively improve the bit error rate (BER) and frame error rate (FER) performance compared with the conventional selection method, especially in high signal noise ratio (SNR) region. Moreover, the algorithm can be used to construct polar codes with any rate through a complete iteration.