A wiretap channel is served as the fundamental model of physical layer security techniques, where the secrecy capacity of the Gaussian wiretap channel is proven to be achieved by Gaussian input. However, there remains a gap between the Gaussian secrecy capacity and the secrecy rate with conventional uniformly distributed discrete constellation input, e.g. amplitude shift keying (ASK) and quadrature amplitude modulation (QAM). In this paper, we propose a probabilistic shaped multilevel polar coding scheme to bridge the gap. Specifically, the input distribution optimization problem for maximizing the secrecy rate with ASK/QAM input is solved. Numerical results show that the resulting sub-optimal solution can still approach the Gaussian secrecy capacity. Then, we investigate the polarization of multilevel polar codes for the asymmetric discrete memoryless wiretap channel, and thus propose a multilevel polar coding scheme integration with probabilistic shaping. It is proved that the scheme can achieve the secrecy capacity of the Gaussian wiretap channel with discrete constellation input, and satisfies the reliability condition and weak security condition. A security-oriented polar code construction method to natively satisfies the leakage-based security condition is also investigated. Simulation results show that the proposed scheme achieves more efficient and secure transmission than the uniform constellation input case over both the Gaussian wiretap channel and the Rayleigh fading wiretap channel.

We present a communication scheme using guessing random additive noise decoding (GRAND) to improve flexibility and reliability of the existing compressed error (CE) framework. The CE framework uses information feedback to construct follow-up transmissions by compressing previous noise realizations, offering high reliability at a code rate close to the forward channel capacity. The channel decoding algorithm GRAND allows us to efficiently maintain this performance in noisy feedback settings by shifting redundancy for forward message protection to the feedback channel. Our scheme, GRAND-CE, is therefore appropriate for cases where forward and feedback channel usage costs are asymmetric, e.g. uplink communications. GRAND-CE offers super-exponential error rate performance as channel use increases, with finite usage of a noisy feedback channel. Unlike the traditional forward error correction model, the receiver performs error correction encoding and the sender handles decoding. We also propose a technique for pipelining sequential transmissions to maintain fixed forward transmission length and good feedback channel coding performance.

We introduce an algorithm for approximating the codebook probability that is compatible with all successive cancellation (SC)-based decoding algorithms, including SC list (SCL) decoding. This approximation is based on an auxiliary distribution that mimics the dynamics of decoding algorithms with an SC decoding schedule. Based on this codebook probability and SCL decoding, we introduce soft-output SCL (SO-SCL) to generate both blockwise and bitwise soft-output (SO). Using that blockwise SO, we first establish that, in terms of both block error rate (BLER) and undetected error rate (UER), SO-SCL decoding of dynamic Reed-Muller (RM) codes significantly outperforms the CRC-concatenated polar codes from 5G New Radio under SCL decoding. Moreover, using SO-SCL, the decoding misdetection rate (MDR) can be constrained to not exceed any predefined value, making it suitable for practical systems. Proposed bitwise SO can be readily generated from blockwise SO via a weighted sum of beliefs that includes a term where SO is weighted by the codebook probability, resulting in a soft-input soft-output (SISO) decoder. Simulation results for SO-SCL iterative decoding of product codes and generalized LDPC (GLDPC) codes, along with information-theoretical analysis, demonstrate significant superiority over existing list-max and list-sum approximations.

We establish that a large, flexible class of long, high redundancy error correcting codes can be efficiently and accurately decoded with guessing random additive noise decoding (GRAND). Performance evaluation demonstrates that it is possible to construct simple product codes with lengths of approximately 200 to 4000 bits and rates between 0.2 and 0.8 that outperform low-density parity-check (LDPC) codes from the 5G New Radio standard in both AWGN and fading channels. The concatenated structure enables many desirable features, including: low-complexity hardware-friendly encoding and decoding; significant flexibility in length and rate through modularity; and high levels of parallelism in encoding and decoding that enable low latency. Central is the development of a method through which any soft-input (SI) GRAND algorithm can provide soft-output (SO) in the form of an accurate a-posteriori estimate of the likelihood that a decoding is correct or, in the case of list decoding, the likelihood that each element of the list is correct. The distinguishing feature of soft-output GRAND (SOGRAND) is the provision of an estimate that the correct decoding has not been found, even when providing a single decoding. Per-block SO can be converted into accurate per-bit SO by a weighted sum that includes a term for the SI. Implementing SOGRAND adds negligible computation and memory to the existing decoding process, and using it results in a practical, low-latency alternative to LDPC codes.

We present a framework that can exploit the tradeoff between the undetected error rate (UER) and block error rate (BLER) of polar-like codes. It is compatible with all successive cancellation (SC)-based decoding methods and relies on a novel approximation that we call codebook probability. This approximation is based on an auxiliary distribution that mimics the dynamics of decoding algorithms following an SC decoding schedule. Simulation results demonstrates that, in the case of SC list (SCL) decoding, the proposed framework outperforms the state-of-art approximations from Forney's generalized decoding rule for polar-like codes with dynamic frozen bits. In addition, dynamic Reed-Muller (RM) codes using the proposed generalized decoding significantly outperform CRC-concatenated polar codes decoded using SCL in both BLER and UER.

A tree search algorithm called successive cancellation ordered search (SCOS) is proposed forGN-coset codes that implements maximum-likelihood (ML) decoding with adaptive complexity for transmission over binary-input AWGN channels. Unlike bit-flip decoders, no outer code is needed to terminate decoding; therefore, SCOS also applies toGN-coset codes modified with dynamic frozen bits. The average complexity is close to that of successive cancellation (SC) decoding at practical frame error rates (FERs) for codes with wide ranges of rate and lengths up to 512 bits, which perform within 0.25 dB or less from the random coding union bound and outperform Reed–Muller codes under ML decoding by up to 0.5 dB. Simulations illustrate simultaneous gains for SCOS over SC-Fano, SC stack (SCS) and SC list (SCL) decoding in FER and the average complexity at various SNR regimes. SCOS is further extended by forcing it to look for candidates satisfying a threshold, thereby outperforming basic SCOS under complexity constraints. The modified SCOS enables strong error-detection capability without the need for an outer code. In particular, the (128, 64) polarization-adjusted convolutional code under modified SCOS provides gains in overall and undetected FER compared to CRC-aided polar codes under SCL/dynamic SC flip decoding at high SNR.

Guessing Random Additive Noise Decoding (GRAND) is a family of hard- and soft-detection error correction decoding algorithms that provide accurate decoding of any moderate redundancy code of any length. Here we establish a method through which any soft-input GRAND algorithm can provide soft output in the form of an accurate a posteriori estimate of the likelihood that a decoding is correct or, in the case of list decoding, the likelihood that the correct decoding is an element of the list. Implementing the method adds negligible additional computation and memory to the existing decoding process. The output permits tuning the balance between undetected errors and block errors for arbitrary moderate redundancy codes including CRCs.

Guessing random additive noise decoding (GRAND) has recently demonstrated maximum-likelihood (ML) decoding performance on efficient, universal silicon realizations. Leveraging input bit-reliability soft information extracted from the channel and noise statistics, GRAND rank-orders and queries noise sequences in non-decreasing likelihood to recover code-words of arbitrary code-book structures. We consider soft-input, soft-output (SISO) GRAND that generates bit-reliability log-likelihood ratios (LLRs) via successive Euclidean-distance computations over a list of noise-recovered words. Noise guessing and list construction follow an ordered reliability bits GRAND (ORBGRAND) mechanism, the guess budget of which controls the performance and complexity trade-offs. The generated LLRs form enhanced a priori information that adapts noise-sequence ordering in a subsequent soft-GRAND iteration. We derive bounds on the achievable rates under per-realization and marginal input soft information and empirically study the achievable rates of SISO-GRAND. We also examine the complexity of the joint data detection and GRAND core, highlighting its superiority to conventional list-based detection schemes. SISO-ORBGRAND can outperform conventional sphere decoding in data detection and LLR generation; the corresponding channel-mismatched rates approximate ML decoding.

In a setting where the forward and feedback channel are noisy BSCs, we show how capacity is nearly achievable in a scheme with only source coding on the forward channel. In representative settings with noisy feedback, GRAND makes the scheme not only possible but practical. The sender transmits uncoded messages, and the receiver provides a noise effect guess as in GRAND, which is channel-encoded and sent to the receiver. With noiseless, finite-length feedback our scheme provides the type of super exponential error behavior associated in previous work with infinite-capacity feedback channels. With noisy feed-back, which is the more common setting in most systems, our scheme permits forward throughput that is effectively the same as in a noiseless feedback case. Moreover, the feedback channel usage remains limited. We propose a target error rate as a useful design parameter.

A complexity-adaptive tree search algorithm is proposed for $G_{N}$-coset codes that implements maximum-likelihood (ML) decoding by using a successive decoding schedule. The average complexity is close to that of the successive cancellation (SC) decoding for practical error rates when applied to polar codes and short Reed-Muller (RM) codes, e.g., block lengths up to N = 128. By modifying the algorithm to limit the worstcase complexity, one obtains a near-ML decoder for longer RM codes and their subcodes. Unlike other bit-flip decoders, no outer code is needed to terminate decoding. The algorithm can thus be applied to modified $G_{N}$-coset code constructions with dynamic frozen bits. One advantage over sequential decoders is that there is no need to optimize a separate parameter.

Polar codes are designed for parallel binary-input additive white Gaussian noise (BiAWGN) channels with an average power constraint. The two main design choices are: the mapping between codeword bits and channels of different quality, and the power allocation under the average power constraint. Information theory suggests to allocate power such that the sum of mutual information (MI) terms is maximized. However, a power allocation specific to polar codes shows significant gains.