Arikan's polar codes are capable of achieving the Shannon's capacity at a low encoding and decoding complexity, while inherently supporting rate adaptation. By virtue of these attractive features, polar codes have provided fierce competition to both the turbo as well as the Low Density Parity Check (LDPC) codes, making its way into the 5G New Radio (NR). Realizing the significance of polar codes, in this paper we provide a comprehensive survey of polar codes, highlighting the major milestones achieved in the last decade. Furthermore, we also provide tutorial insights into the polar encoder, decoders as well as the code construction methods. We also extend our discussions to quantum polar codes with an emphasis on the underlying quantum-to-classical isomorphism and the syndromebased quantum polar codes.
This paper provides a detailed tutorial on the Cyclic Redundancy Check (CRC)-aided logarithmic successive cancellation stack (Log-SCS) algorithm. We apply these algorithms for the ultrareliable decoding of polar codes, which has relevance for the control channels of the ultra-reliable low latency communication version of the third generation partnership project (3GPP) New Radio (NR). During the exploitation of the CRC codes to improve the error correction performance, we propose a novel technique which limits the number of CRC checks performed, in order to maintain a consistent error detection performance. In addition, we propose a pair of techniques for further improving the performance of the Log-SCS polar decoder. We demonstrate that the proposed S = 128 Improved Log-SCS decoder achieves a similar error correction capability as a logarithmic successive cancellation list (Log-SCL) decoder having a list size of L = 128 across the full range of block lengths supported by the 3GPP NR physical uplink control channel. This is achieved without increasing its memory requirement, while dramatically reducing its complexity, which becomes up to seven times lower than that of a L = 8 Log-SCL decoder.
Soft-output (SO) decoding is proposed for the Logarithmic Successive Cancellation List (Log-SCL) polar decoder for the first time, by exploiting the left-to-right propagation of the Belief Propagation (BP) decoder, which opens new avenues for its employment in powerful turbo-receivers. In the case of decoding a half-rate polar code having a block length of 1024 bits, the proposed soft list polar decoder achieves a 1.5 dB Block Error Ratio (BLER) performance gain, 50% latency improvement and 26% complexity reduction, compared to the state-of-the-art SO Soft Cancellation (SCAN) polar decoder in a polar-coded Multiple-Input Multiple-Output (MIMO) system. Furthermore, we conceive a Memory-Efficient (ME) soft list polar decoder, which requires only 16% of the soft list polar decoder's memory, at the cost of slightly increased latency and complexity. I. INTRODUCTION Polar codes [1] have been selected for protecting the 5G New Radio (NR) control channels, as a benefit of their superior Block Error Ratio (BLER) performance at short block lengths [2]. Therefore, both Hard-Output (HO) and Soft-Output (SO) polar decoders have been investigated in the literature [1, 3-8]. The soft-input HO (SIHO) polar decoders typically rely on the original Successive Cancellation (SC) decoder proposed in [1] or sphere decoder of [9], whereas SISO may be produced by the Belief Propagation (BP) algorithm [3, 10]. The SC decoder is capable of approaching the Binary-Input Discrete Memoryless Channel (B-DMC) capacity for infinite block lengths [1] where the channel is sufficiently polarized, but exhibits poor BLER performance for realistic finite block lengths due to the associated error propagation. Considering a list of candidate bit sequences and operating on the basis of Logarithmic Likelihood Ratios (LLRs), the Logarithmic Successive Cancellation List (Log-SCL) polar decoder, developed from the SCL decoder of [11], improves the error-correction performance of the SC decoder in the case of practical block lengths [4]. The software and hardware implementations of the Log-SCL decoder have been investigated in [12-15], which have a low memory requirement and a low decoding latency. However, the Log-SCL decoder is not a SISO scheme, and hence fails to exploit the iterative LLR updates gleaned from the detector of a turbo-receiver [5, 16].
Since their inception in 2008, polar codes have been shown to offer near-capacity error correction performance across a wide range of block lengths and coding rates. Owing to this, polar codes have been selected to provide channel coding in the control channels of Third Generation Partnership Project (3GPP) New Radio (NR). The operation of the 3GPP NR polar codes is specified in the 3GPP standard TS 38.212, together with schemes for code block segmentation, Cyclic Redundancy Check (CRC) attachment, CRC scrambling, CRC interleaving, frozen and parity check bit insertion, sub-block interleaving, bit selection, channel interleaving and code block concatenation. The configuration of these components is different for the uplink, broadcast and downlink control channels. However, the lack of visualisations and diagrammatic explanations in TS 38.212 limits the accessibility of the standard to new readers. This motivates the aims of the paper, which provides detailed tutorials on the operation and motivation of the components of the 3GPP NR polar codes, as well as surveys of the 3GPP discussions that led to their specification. Furthermore, we comprehensively characterize the error correction and error detection performance of the 3GPP NR polar codes in the uplink, broadcast and downlink control channels.
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