Rate-compatible Polar Codes Research Articles

Rate-Compatible Punctured Polar Codes

In this work, we propose a novel puncturing scheme for rate-compatible polar codes of finite lengths. By analyzing the log-likelihood ratio (LLR) message propagation from the channel outputs to the information bits, we show that there exist certain paths, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">direct paths</i> through which the reliabilities of LLR values at the information bits are strongly influenced by the channel outputs. For each information bit, we define a metric called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">reliability score</i> which measures the reliability of LLR value at the information bit built up through the direct paths. Then, we propose a rate-compatible puncturing scheme which decides punctured bits in such a way as to maximize the minimum reliability score. Simulation results show that the proposed puncturing scheme significantly outperforms existing schemes in terms of error-rate performance.

Polar Codes: Encoding/Decoding and Rate-Compatible Jointly Design for HARQ System

Polar coding are the first class of provable capacity-achieving coding techniques for a wide range of channels. With an ideal recursive structure and many elegant mathematical properties, polar codes are inherently implemented with low complexity encoding and decoding algorithms. Since the block length of the original polar construction is limited to powers of two, rate-compatible polar codes (RCPC) are presented to meet the flexible length/rate transmission requirements in practice. The RCPC codes are well-conditioned to combine with the hybrid automatic repeat request (HARQ) system, providing high throughput efficiency and such RCPC-HAPQ scheme is commonly used in delay-insensitive communication system. This paper first gives a survey of both the classical and state-of-the-art encoding/decoding algorithms for polar codes. Then the RCPC construction methods are discussed, including the puncturing, shortening, multi-kernel construction, etc. Finally, we investigate several RCPC-HARQ jointly design systems and discuss their encoding gain and re-transmission diversity gain.

Rate‐compatible systematic polar codes

Puncturing and shortening are two common ways to achieve rate-compatible non-systematic polar codes (NSPCs). Systematic polar codes (SPCs) have been shown to outperform NSPCs with the same encoding and decoding complexity. However, rate-compatible SPCs have never been comprehensively studied in previous work. In this paper, two rate-compatible algorithms for SPCs are first proposed: uniform puncturing (UP) algorithm and uniform shortening (US) algorithm, which are referred to as SPC-UP and SPC-US, respectively. In order to effectively estimate the maximum likelihood decoding performance of punctured and shortened polar codes, subsequently, a distance spectrum calculation algorithm based on successive cancellation list (SCL) decoder for rate-compatible polar codes is proposed. Simulation results show that rate-compatible SPCs yield better bit error rate performance than rate-compatible NSPCs while they have the same frame error rate performance under different code rates and decoding algorithms. Eventually, union bounds that are obtained by the distance spectrum to provide the theoretical explanation for the superiority of rate-compatible SPCs are utilised.

Design of Rate-Compatible Polar Codes Based on Non-Uniform Channel Polarization

In this article we propose a technique for polar codes (PC) construction for any code length. By default, PC construction is limited to code length proportional to the power of two. To construction the code length arbitrary, puncturing, shortening and extension techniques must be applied. However, performance is degraded with the use of these techniques. Other ways to design polar codes with arbitrary code length but which have encoding and decoding with higher complexity such as multi-kernel, concatenated codes and specific constructions for belief propagation (BP) or successive cancellation list (SCL) decoding. The polarization theory is generalized for non-uniform channels (NUC) and with this approach we can construction rate-compatible PC and variable code length. We developed an implementation algorithm based on the of PC construction by Gaussian approximation (NUPGA). In a scenario where the transmission is over an additive white Gaussian noise (AWGN) channel and under successive cancellation (SC) decoding, the PC construction of arbitrary code length can be implemented with NUPGA. With NUPGA we re-polarize the projected synthetic channels by choosing more efficiently the positions of the information bits. In addition, we present a generalization of the Gaussian approximation (GA) for the polarization and re-polarization processes and an extension technique for PC. The PC construction based on NUPGA present better performance than the existing techniques as shown in the simulations of this work.

On Training Neural Network Decoders of Rate Compatible Polar Codes via Transfer Learning.

Neural network decoders (NNDs) for rate-compatible polar codes are studied in this paper. We consider a family of rate-compatible polar codes which are constructed from a single polar coding sequence as defined by 5G new radios. We propose a transfer learning technique for training multiple NNDs of the rate-compatible polar codes utilizing their inclusion property. The trained NND for a low rate code is taken as the initial state of NND training for the next smallest rate code. The proposed method provides quicker training as compared to separate learning of the NNDs according to numerical results. We additionally show that an underfitting problem of NND training due to low model complexity can be solved by transfer learning techniques.

Syndrome-Coupled Rate-Compatible Error-Correcting Codes: Theory and Application

Rate-compatible error-correcting codes (ECCs), which consist of a set of extended codes, are of practical interest in both wireless communications and data storage. In this work, we first study the lower bounds for rate-compatible ECCs, thus proving the existence of good rate-compatible codes. Then, we propose a general framework for constructing rate-compatible ECCs based on cosets and syndromes of a set of nested linear codes. We evaluate our construction from two points of view. From a combinatorial perspective, we show that we can construct rate-compatible codes with increasing minimum distances, and we discuss decoding algorithms and correctable patterns of errors and erasures. From a probabilistic point of view, we prove that we are able to construct capacity-achieving rate-compatible codes, generalizing a recent construction of capacity-achieving rate-compatible polar codes. Applications of rate-compatible codes to data storage are considered. We design two-level rate-compatible codes based on Bose-Chaudhuri-Hocquenghem (BCH) and low-density parity-check (LDPC) codes which are two popular codes widely used in the data storage industry, and then we evaluate the performance of these codes in multi-level cell (MLC) flash memories. We also examine code performance on binary and $q$ -ary symmetric channels. Finally, we briefly discuss two variations of our main construction and their relative performance.

A Mapping Shortening Algorithm for Polar Codes

Shortening is a common way to achieve rate-compatible polar codes. The existing shortening algorithms select shortened bits merely according to the structure of the generator matrix in order to make them known by the receiver. In this paper, we take into account the effect that the shortening has on the capacity of split channels and propose a shortening capacity mapping criterion. Based on the proposed mapping criterion, a mapping shortening (MS) algorithm is proposed. We theoretically prove that the MS algorithm can ensure that the proposed mapping criterion can be adopted reasonably and the shortened bits can be known by the receiver. In addition, the MS algorithm is proved to have the same order of complexity as existing shortening algorithms. What's more, we demonstrate the superiority of the MS algorithm over existing shortened algorithms from the perspective of channel capacity. Finally, the simulation results show that the MS algorithm has a significant advantage over existing shortening algorithms for the bit error rate (BER) and frame error rate (FER) performance under high code rates.

Rate-Compatible Polar Codes Based on Polarization-Driven Shortening

This letter presents a polarization-driven (PD) shortening technique for the design of rate-compatible polar codes. The proposed shortening strategy consists of reducing the generator matrix by relating its row index with the channel polarization index. We assume that the shortened bits are known by both the encoder and the decoder and employ successive cancellation for decoding the shortened codes constructed by the proposed PD technique. A performance analysis is then carried out based on the spectrum distance. Simulations show that the proposed PD-based shortened polar codes outperform existing shortened polar codes.

An Adaptive IR-HARQ Scheme for Polar Codes by Polarizing Matrix Extension

In this letter, we construct rate compatible polar codes via polarizing matrix extension, to support efficient adaptive incremental redundancy hybrid automatic repeat request (IR-HARQ) protocol. We show that an arbitrary number of incremental coded bits can be generated by extending the polarizing matrix, such that multiple retransmissions are aggregated to produce a longer polar code with extra coding gain. This coding gain is maximized by selecting the most reliable bit channels in the extended codes, rather than the shorter codes, as information bits. Naturally, some information bits are transferred from the shorter part to the extended part for higher bit channel reliability. One-to-one parity-checking between the two parts lends itself well to the above purpose during the construction of the longer code. Simulation results demonstrate consistent coding gain over the chase combining HARQ scheme.

Capacity-Achieving Rate-Compatible Polar Codes

We present a method of constructing rate-compatible polar codes that are capacity-achieving with low-complexity sequential decoders. The proposed code construction allows for incremental retransmissions at different rates in order to adapt to channel conditions. The main idea of the construction exploits the common characteristics of polar codes optimized for a sequence of successively degraded channels. The proposed approach allows for an optimized polar code to be used at every transmission thereby achieving capacity.

Circular Buffer Rate-Matched Polar Codes

A practical rate-matching system for constructing rate-compatible polar codes is proposed. The proposed polar code circular buffer rate-matching is suitable for transmissions on communication channels that support hybrid automatic repeat request (HARQ) communications, as well as for flexible resource-element rate-matching on single transmission channels. Our proposed circular buffer rate matching scheme also incorporates a bit-mapping scheme for transmission on bit-interleaved coded modulation (BICM) channels using higher order modulations. An interleaver is derived from a puncturing order obtained with a low complexity progressive puncturing search algorithm on a base code of short length, and has the flexibility to achieve any desired rate at the desired code length, through puncturing or repetition. The rate-matching scheme is implied by a two-stage polarization, for transmission at any desired code length, code rate, and modulation order, and is shown to achieve the symmetric capacity of BICM channels. Numerical results on AWGN and fast fading channels show that the rate-matched polar codes have a competitive performance when compared to the spatially-coupled quasi-cyclic LDPC codes or LTE turbo codes, while having similar rate-dematching storage and computational complexities.

A Hybrid ARQ Scheme Based on Polar Codes

A hybrid automatic repeat request (HARQ) scheme based on a novel class of rate-compatible polar (\mbox{RCP}) codes are proposed. The RCP codes are constructed by performing punctures and repetitions on the conventional polar codes. Simulation results over binary-input additive white Gaussian noise channels (BAWGNCs) show that, using a low-complexity successive cancellation (SC) decoder, the proposed HARQ scheme performs as well as the existing schemes based on turbo codes and low-density parity-check (LDPC) codes. The proposed transmission scheme is only about 1.0-1.5dB away from the channel capacity with the information block length of 1024 bits.