This paper focuses on the frozen set design for precoded polar codes decoded by the successive cancellation list (SCL) algorithm. We propose a novel frozen set design method, whose computational complexity is low due to the use of analytical bounds and constrained frozen set structure. We derive new bounds based on the recently published complexity analysis of SCL decoding with near maximum-likelihood (ML) performance. To predict the ML performance, we employ the state-of-the-art bounds relying on the code weight distribution. The bounds and constrained frozen set structure are incorporated into the genetic algorithm to generate optimized frozen sets with low complexity. Our simulation results show that the constructed precoded polar codes of length 512 have a superior frame error rate (FER) performance compared to the state-of-the-art codes under SCL decoding with various list sizes.

This paper focuses on the design of high-performance polar codes with dynamic frozen bits that can be compactly specified. We split the code design problem into the frozen set design and the frozen bit expression design problems. To solve the first problem, we analyze the connection between the code minimum distance and the frozen set. This analysis leads to a novel frozen set structure that ensures a low frame error rate (FER) under successive cancellation list (SCL) decoding. Given a bit-channel reliability sequence, our frozen set structure reduces the problem of frozen set design to that of selecting three integer numbers. We develop a reinforcement learning technique to find the values of these integer numbers minimizing the FER under SCL decoding. We then propose a simple deterministic method producing efficient expressions for dynamic frozen bits. Simulation results show that the proposed compactly-specified polar codes of lengths 512 to 4096 outperform the state-of-the-art polar code constructions under SCL decoding in the high SNR regime.

In this paper, we propose a novel artificial intelligence (AI) based adaptive polar coding scheme that adapts to various channel conditions and quality of service requirements. To ensure tight adaptation, we develop a new AI-based performance prediction framework for the precoded polar codes under the successive cancellation list (SCL) decoder. This AI-based framework relies on a neural network and recent advancements in the analysis of precoded polar codes, SCL and SC decoders. Then we apply the proposed framework to optimise precoded polar codes for various target frame error rates (FER), signal-to-noise ratios (SNR) and decoding list sizes$L$, where the code length is fixed to a power of two, but the code rate may vary. We predict the throughput and maximise it over the code rates with bit-level granularity. The proposed approach paves the way towards online adaptive polar coding with high error-correction capability. The constructed codes can be compactly specified using the reliability sequence from the 5G New Radio standard and a single parameter whose value is specific to each code. The simulation results show that the proposed codes outperform 5G polar codes with CRC11 under SCL decoding with various$L$.

Multiuser massive multiple-input multiple-output (MU-MIMO) systems can be used to meet high throughput requirements of 5G and beyond networks. In an uplink MU-MIMO system, a base station is serving a large number of users, leading to a strong multi-user interference (MUI). Designing a high performance detector in the presence of a strong MUI is a challenging problem. This work proposes a novel detector based on the concepts of expectation propagation (EP) and graph neural network, referred to as the GEPNet detector, addressing the limitation of the independent Gaussian approximation in EP. The simulation results show that the proposed GEPNet detector significantly outperforms the state-of-the-art MU-MIMO detectors in strong MUI scenarios with equal number of transmit and receive antennas.

Multi-user multiple-input multiple-output (MU-MIMO) systems can be used to meet high throughput requirements of 5G and beyond networks. A base station serves many users in an uplink MU-MIMO system, leading to a substantial multi-user interference (MUI). Designing a high-performance detector for dealing with a strong MUI is challenging. This paper analyses the performance degradation caused by the posterior distribution approximation used in the state-of-the-art message passing (MP) detectors in the presence of high MUI. We develop a graph neural network based framework to fine-tune the MP detectors’ cavity distributions and thus improve the posterior distribution approximation in the MP detectors. We then propose two novel neural network based detectors which rely on the expectation propagation (EP) and Bayesian parallel interference cancellation (BPIC), referred to as the GEPNet and GPICNet detectors, respectively. The GEPNet detector maximizes detection performance, while GPICNet detector balances the performance and complexity. We provide proof of the permutation equivariance property, allowing the detectors to be trained only once, even in the systems with dynamic changes of the number of users. The simulation results show that the proposed GEPNet detector performance approaches maximum likelihood performance in various configurations and GPICNet detector doubles the multiplexing gain of BPIC detector.

The problem of computing the Hamming weight distribution of linear codes is considered in this paper. A novel method to enumerate all codewords up to a certain Hamming weight for binary linear block codes in general and in particular for the punctured, shortened, precoded polar codes is introduced. The proposed approach performs a recursive decomposition of the codes using construction X4 that is typically used to combine codes of different lengths. This allows to enumerate the low-weight codewords of the overall code as combinations of the low-weight codewords of the component codes. Numerical results show that the proposed approach can efficiently compute the exact partial weight distribution of the 5G New Radio punctured/shortened polar codes with CRC11 and pure polar codes. In the former and latter cases, the low-weight codeword number is up to 106 and 108, respectively. Besides, randomly punctured and shortened polar codes and randomly precoded polar codes are also considered. To the best of the authors’ knowledge, this is the first method able to solve these problems.

Cell-free (CF) massive multiple-input multiple-output (M-MIMO) technology plays a prominent role in the beyond fifth-generation (5G) networks. However, designing a high performance CF M-MIMO detector is a challenging task due to the presence of pilot contamination which appears when the number of pilot sequences is smaller than the number of users. This work proposes a CF M-MIMO detector referred to as CF expectation propagation (CF-EP) that incorporates the pilot contamination when calculating the posterior belief. The simulation results show that the proposed detector achieves significant improvements in terms of the bit-error rate and sum spectral efficiency performances as compared to the ones of the state-of-the-art CF detectors.

The stringent requirements on reliability and processing delay in the fifth-generation (5G) cellular networks introduce considerable challenges in the design of massive multiple-input-multiple-output (M-MIMO) receivers. The two main components of an M-MIMO receiver are a detector and a decoder. To improve the trade-off between reliability and complexity, a Bayesian concept has been considered as a promising approach that enhances classical detectors, e.g. minimum-mean-square-error detector. This work proposes an iterative M-MIMO detector based on a Bayesian framework, a parallel interference cancellation scheme, and a decision statistics combining concept. We then develop a high performance M-MIMO receiver, integrating the proposed detector with a low complexity sequential decoding for polar codes. Simulation results of the proposed detector show a significant performance gain compared to other low complexity detectors. Furthermore, the proposed M-MIMO receiver with sequential decoding ensures one order magnitude lower complexity compared to a receiver with stack successive cancellation decoding for polar codes from the 5G New Radio standard.

A novel method to recursively construct a set of precoded polar codes of various rates and short-to-moderate lengths is presented. The proposed code design method minimizes the successive cancellation (SC) decoding error probability estimate under three constraints. The first constraint is the minimum distance requirement to improve the maximum-likelihood (ML) performance of the resulting code and therefore the performance under the SC list (SCL) decoding. The other two constraints introduce preselected supercode and subcode, where the supercode ensures fast computation of the minimum distance and the subcode ensures reduction of the search space size. The supercode is given by the Plotkin sum of shorter codes, which are nested to simplify computation of low-weight codewords. These low-weight codewords are needed to satisfy the minimum distance constraint. The simulation results indicate that the proposed precoded polar codes of lengths 128 and 256 provide a better frame error rate (FER) than polar codes with CRC and e-BCH polar subcodes under the SCL decoding algorithm with the list size$8-128$.

The problem of designing polar-like codes for successive cancellation list (SCL) decoding algorithm is considered. A novel code design algorithm aimed to minimize both successive cancellation (SC) and maximum-likelihood (ML) decoding error probabilities is introduced. The algorithm performs optimization of a precoding matrix for the polarization transformation matrix to guarantee preselected minimum distance and the lowest possible SC decoding error probability to the resulting code. Constructed precoded polar codes are decoded as polar codes with dynamic frozen bits. Numerical results show that the proposed codes of lengths 32 and 64 significantly reduce the frame error rate (FER) compared to the original polar codes under the SCL decoding starting with the list size 4. The gain increases with the list size. The proposed codes have a much lower SC decoding error probability than extended Bose-Chaudhuri-Hocquenghem (e-BCH) codes and their polar subcodes without sacrificing the ML performance. In case of the code length 64, the constructed precoded polar codes demonstrate a FER reduction compared to the polar subcodes of e-BCH codes under the SCL decoding with the list size up to 16 and demonstrate a comparable FER starting with the list size 32.

An extension of polar codes is proposed, which allows some of the frozen symbols, called dynamic frozen symbols, to be data-dependent. A construction of polar codes with dynamic frozen symbols, being subcodes of extended BCH codes, is proposed. The proposed codes have higher minimum distance than classical polar codes, but still can be efficiently decoded using the successive cancellation algorithm and its extensions. The codes with Arikan, extended BCH and Reed-Solomon kernel are considered. The proposed codes are shown to outperform LDPC and turbo codes, as well as polar codes with CRC.

A low-complexity systematic encoding algorithm for polar codes with Reed-Solomon (RS) kernel is presented. The proposed method relies on fast Fourier transform-based RS encoding techniques. An application of polar codes in storage systems is considered.

An optimization algorithm for finding a shortening pattern and a set of frozen symbols for polar codes is proposed. The structure of polar codes is exploited to eliminate many equivalent shortening patterns, thus reducing the search space. A reduced-complexity suboptimal algorithm is proposed for finding shortening patterns for long polar codes. Shortened codes obtained with the proposed method are shown to outperform low-density parity-check (LDPC) codes.

The problem of efficient soft-decision decoding of polar codes with any binary kernel is considered. The proposed approach represents a generalization of the sequential decoding algorithm introduced recently for the case of polar codes with Arikan kernel. Numeric results show that the proposed algorithm enables near-ML decoding of polar codes with BCH kernel.

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 problem of construction of binary polar codes with high-dimensional kernels is considered. A novel method for computing the erasure probability in the bit subchannels induced by the polarization kernel is proposed. The codes obtained using the proposed method outperform those based on the Arikan kernel.