Fair-density parity-check (FDPC) codes have been recently introduced demonstrating improved performance compared to low-density parity-check (LDPC) codes standardized in 5G systems particularly in high-rate regimes. In this paper, we introduce a layered normalized min-sum (LNMS) message-passing decoding algorithm for the FDPC codes. We also introduce a syndrome-guided bit flipping (SGBF) method to enhance the error-correction performance of our proposed decoder. The LNMS decoder leverages conflict graph coloring for efficient layered scheduling, enabling faster convergence by grouping non-conflicting check nodes and updating variable nodes immediately after each layer. In the event of decoding failure, the SGBF method is activated, utilizing a novel reliability metric that combines log-likelihood ratio (LLR) magnitudes and syndrome-derived error counts to identify the least reliable bits. A set of candidate sequences is then generated by performing single-bit flips at these positions, with each candidate re-decoded via LNMS. The optimal candidate is selected based on the minimum syndrome weight. Extensive simulation results demonstrate the superiority of the proposed decoder. Numerical simulations on FDPC$(256,192)$ code with a bit-flipping set size of $T = 128$ and a maximum of $5$ iterations demonstrate that the proposed decoder achieves approximately a $0.5\,\mathrm{dB}$ coding gain over standalone LNMS decoding at a frame error rate (FER) of $10^{-3}$, while providing coding gains of $0.75-1.5\,\mathrm{dB}$ over other state-of-the-art codes including polar codes and 5G-LDPC codes at the same length and rate and also under belief propagation decoding.

In this paper, we propose a method for obtaining metric functions for each depth of the channel polarization tree through a process that we call polarization of the metric function. One of the major advantages of the proposed metric function is that it can be utilized in fast successive cancellation-based (FSC) and SC list-based (FSCL) decoders, i.e., decoders that opt to skip the so-called rate-1 and rate-0 nodes in the binary tree representation for significantly more efficient implementation. Furthermore, we relate the average and variance values of the polarized metric function of FSC-based decoders to the polarized channel capacity and polarized varentropy. By leveraging this observations, we introduce a pruning technique that keeps only the paths in the FSCL decoder whose metric values are close to the average value. As a result, our proposed technique significantly reduces the number of required sorting operations for FSCL-based decoding algorithms. For instance, for a highrate PAC (128,99) code, SCL decoding with a list size of 32 achieves error-correction performance comparable to the Fano algorithm. FSCL decoding requires visiting only 28 nodes of the polarization tree, significantly fewer than the 254 nodes required for conventional SCL decoding. Additionally, our method reduces the number of sorting operations by a factor of 3, further decreasing latency and complexity.

In this paper, we explore the design and analysis of regular bipartite graphs motivated by their application in lowdensity parity-check (LDPC) codes specifically with constrained girth and in the high-rate regime. We focus on the relation between the girth of the graph, and the size of the sets of variable and check nodes. We derive bounds on the size of the vertices in regular bipartite graphs, showing how the required number of check nodes grows with respect to the number of variable nodes as girth grows large. Furthermore, we present two constructions for bipartite graphs with girth$\mathcal{G}=8$; one based on a greedy construction of ($w_{c}, w_{r}$) -regular graphs, and another based on semi-regular graphs which have uniform column weight distribution with a sublinear number of check nodes. The second construction leverages sequences of integers without any length- 3 arithmetic progression and is asymptotically optimal while maintaining a girth of 8. Also, both constructions can offer sparse parity-check matrices for high-rate codes with medium-to-large block lengths. Our results solely focus on the graph-theoretic problem but can potentially contribute to the ongoing effort to design LDPC codes with high girth and minimum distance, specifically in high code rates.

Abstract Precision Time Protocol, PTP, is one of the most accurate protocols for maintaining clock synchronization in industrial and distributed systems and networks. Besides, profiteering or malicious intentions of the network attackers have led to various types of cyber-attacks on different network parts. Time-Delay Attack, TDA, is a well-known network attack that disrupts synchronization. This paper studies a new TDA detection algorithm to maintain and secure the benefits of PTP in industrial networks. We show that through adding new efficient paths and innovative computations based on End-to-End computing, the reliability and security of PTP increase significantly while the incurred traffic overhead remains negligible. Mathematical analysis and simulations confirm the effectiveness and advantages of the proposed algorithm, especially in case of the variety of attacks detected in the network and the location of their occurrence.

In this paper, we propose a method to obtain the optimal metric function at each depth of the polarization tree through a process we callpolarizationof the metric function. This polarization process generates an optimal metric at intermediate levels of the polarization tree, which can be applied infastsuccessive-cancellation-based (FSC) and SC list-based (FSCL) decoders—decoders that partially explore the binary tree representation. We prove that at each step of the polarization tree, the expected value of the metric function random variable is the mutual information of the corresponding channel, while its variance equals the varentropy of the channel—two parameters that are particularly relevant in finite block-length regimes. Additionally, we show that after polarization, the variances of the bit metrics approach zero for binary-input discrete memoryless channels (BI-DMCs). Moreover, we provide an estimate for calculating the variance of the binary-input additive white Gaussian noise (BI-AWGN) channel. We introduce a list-pruning strategy for FSCL decoding that retains only the paths whose metric values are close to the average. As a result, our method significantly reduces the number of required sorting operations in FSCL-based decoding algorithms. We also derive an upper bound, as a function of the polarized channel varentropy, on the probability that the distance between a bit-metric random variable and the bit-channel mutual information exceeds a given threshold. Leveraging this result, we further propose a varentropy-based list-pruning strategy for the SCL (VPSCL) decoding algorithm that adapts to the varentropy of the corresponding bit-channel. Our proposed pruning strategy also benefits stack decoding (VPStack) by discarding partial paths and avoiding unnecessary extensions.

Recently, a novel variation of polar codes known as polarization-adjusted convolutional (PAC) codes has been introduced by Arıkan. These codes significantly outperform conventional polar and convolutional codes, particularly for short codeword lengths, and are shown to operate very close to the optimal bounds. It has also been shown that if the rate profile of PAC codes does not adhere to certain polarized cutoff rate constraints, the computation complexity for their sequential decoding grows exponentially. In this paper, we address the converse problem, demonstrating that if the rate profile of a PAC code follows the polarized cutoff rate constraints, the required computations for its sequential decoding can be bounded with a distribution that follows a Pareto distribution. This serves as a guideline for the rate-profile design of PAC codes. For a high-rate PAC (1024,899) code, simulation results show that the PAC code with Fano decoder, when constructed based on the polarized cutoff rate constraints, achieves a coding gain of more than 0.75 dB at a frame error rate (FER) of 10−5compared to the state-of-the-art 5G polar and LDPC codes.

In this paper, we introduce a novel rate-profile design based on search-constrained optimization techniques to assess the performance of polarization-adjusted convolutional (PAC) codes under Fano (sequential) decoding. The results demonstrate that the optimized PAC code offers much reduced computational complexity compared to a construction based on a conventional genetic algorithm without a loss in error-correction performance. We propose an adaptive successive cancellation list decoding algorithm as the fitness function of our algorithm to determine the weight distribution of the rate profiles. The simulation results indicate that, for a PAC(256, 128) code, only 8% of the population requires that their fitness function be evaluated with a large list size. This represents an improvement of almost 92% over a conventional evolutionary algorithm. For a PAC(64, 32) code, this improvement is about 99%. We also consider high-rate PAC(128, 105) and PAC(64, 51) codes, showing superior performance compared to other existing algorithms.

Despite the extreme error-correction performance, the amount of computation of sequential decoding of the polarization-adjusted convolutional (PAC) codes is random. In sequential decoding of convolutional codes, the cutoff rate denotes the region between rates whose average computational complexity of decoding is finite and those which is infinite. In this paper, by benefiting from the polarization and guessing techniques, we prove that the required computation in sequential decoding of pre-transformed polar codes polarizes, and this polarization determines which set of bit positions within the rate profile may result in high computational complexity. Based on this, we propose a technique for taming the Reed-Muller (RM) rate-profile construction, and the performance results demonstrate that the error-correction performance of the PAC codes can achieve the theoretical bounds using the tamed-RM rate-profile construction and requires a significantly lower computational complexity than the RM rate-profile construction.

Sorting operation is one of the main bottlenecks for the successive-cancellation list (SCL) decoding. This paper introduces an improvement to the SCL decoding for polar and pre-transformed polar codes that reduces the number of sorting operations without visible degradation in the code’s error-correction performance. In an SCL decoding with an optimum metric function we show that, on average, the correct branch’s bit-metric value must be equal to the bit-channel capacity, and on the other hand, the average bit-metric value of a wrong branch can be at most zero. This implies that a wrong path’s partial path metric value deviates from the bit-channel capacity’s partial summation. For relatively reliable bit-channels, the bit metric for a wrong branch becomes very large negative number, which enables us to detect and prune such paths. We prove that, for a threshold lower than the bit-channel cutoff rate, the probability of pruning the correct path decreases exponentially by the given threshold. Based on these findings, we presented a pruning technique, and the experimental results demonstrate a substantial decrease in the amount of sorting procedures required for SCL decoding. In the stack algorithm, a similar technique is used to significantly reduce the average number of paths in the stack.

In this paper, we present a sequential decoding metric function, which leads to significantly improved computational complexity while maintaining the superiority of polarization-adjusted convolutional (PAC) codes’ error-correction performance. With the proposed metric function, the PAC codes’ decoding computational complexity is comparable to the computational complexity of sequential decoding of conventional convolutional codes (CCs). Moreover, simulation results show an improvement in the error-correction performance of low rate PAC codes when using the proposed metric function. Simulation results also show that using the proposed metric, the upper bound on the PAC codes’ computational complexity has a Pareto distribution. To reduce the worst-case latency of PAC sequential decoder, we limit the number of searches performed by sequential decoder. The results show that for PAC codes of length 128, search-limited sequential decoding can achieve an error-correction performance close to the error-correction performance of polar codes with successive cancellation list decoding with list size 64 and CRC length 11 with considerably less computational complexity.