This paper introduces an efficient algorithm based on the Parity-Consistent Decomposition (PCD) method to determine the WD of pre-transformed polar codes. First, to address the bit dependencies introduced by the pre-transformation matrix, we propose an iterative algorithm to construct an \emph{Expanded Information Set}. By expanding the information bits within this set into 0s and 1s, we eliminate the correlations among information bits, thereby enabling the recursive calculation of the Hamming weight distribution using the \emph{PCD method}. Second, to further reduce computational complexity, we establish the theory of equivalence classes for pre-transformed polar codes. Codes within the same equivalence class share an identical weight distribution but correspond to different \emph{Expanded Information Set} sizes. By selecting the pre-transformation matrix that minimizes the \emph{Expanded Information Set} size within an equivalence class, we optimize the computation process. Numerical results demonstrate that the proposed method significantly reduces computational complexity compared to existing deterministic algorithms.

This paper introduces a novel parity-consistent decomposition (PCD) method for accurately determining the weight distribution (WD) of polar codes. In the PCD method, polar codes are represented as a union of a new type of polar cosets, characterized by the property that any two adjacent bits in the uncoded sequence are either both information bits or both frozen bits. This representation reduces the number of polar cosets compared to the existing automorphism-based (AUT) method. We introduce a recursive decomposition method for this new polar coset, which facilitates efficient recursive computation of the WD of polar codes. The PCD method is organized within a tree structure, which facilitates the evaluation of its computational complexity. We then investigate the transitivity of the action of lower triangular affine (LTA) transformations on a code set composed of the new polar cosets. This transitivity property enables a further reduction in the number of cosets that need to be computed. Furthermore, we propose a pruning algorithm to determine the partial weight distribution (PWD) of polar codes based on the tree structure of the PCD method. Given an upper bound on the Hamming weight for the PWD, we update the Hamming weight threshold for each code in the tree and remove the codes whose minimum Hamming weights exceed their corresponding thresholds. Numerical results demonstrate the superiority of the PCD method over existing methods.

This paper introduces a novel parity-consistent decomposition (PCD) algorithm for accurately determining the weight distribution (WD) of polar codes. In the PCD algorithm, polar codes are represented as a union of a new type of polar cosets, which avoids fixing two adjacent information bits as 0 and 1. This representation reduces the number of polar cosets compared to the existing automorphism-based algorithm. Furthermore, we introduce a recursive decomposition method for this new polar coset, which facilitates efficient recursive computation of the WD of polar codes. The PCD algorithm is organized within a tree-like structure, which facilitates the evaluation of its computational complexity. Comparative analysis with the automorphism-based method demonstrates that the PCD algorithm significantly reduces computational complexity.

Recommender systems are among the most impactful AI applications, interacting with billions of users every day, guiding them to relevant products, services, or information tailored to their preferences.However, the research and development of recommender systems are hindered by existing datasets that fail to capture realistic user behaviors and inconsistent evaluation settings that lead to ambiguous conclusions.This paper introduces the \textbf{O}pen \textbf{R}ecommendation \textbf{B}enchmark for Reproducible Research with H\textbf{I}dden \textbf{T}ests (\textbf{ORBIT}), a unified benchmark for consistent and realistic evaluation of recommendation models. ORBIT offers a standardized evaluation framework of public datasets with reproducible splits and transparent settings for its public leaderboard. Additionally, ORBIT introduces a new webpage recommendation task, ClueWeb-Reco, featuring web browsing sequences from 87 million public, high-quality webpages. ClueWeb-Reco is a synthetic dataset derived from real, user-consented, and privacy-guaranteed browsing data. It aligns with modern recommendation scenarios and is reserved as the hidden test part of our leaderboard to challenge recommendation models' generalization ability. ORBIT measures 12 representative recommendation models on its public benchmark and introduces a prompted LLM baseline on the ClueWeb-Reco hidden test.Our benchmark results reflect general improvements of recommender systems on the public datasets, with variable individual performances.The results on the hidden test reveal the limitations of existing approaches in large-scale webpage recommendation and highlight the potential for improvements with LLM integrations.ORBIT benchmark, leaderboard, and codebase are available at \url{https://www.open-reco-bench.ai}.

Rate-splitting multiple access (RSMA) is a flexible multiple access scheme that leverages the advantages of superposition coding to enhance user throughput, making it particularly advantageous for Internet of Things (IoT) scenarios. In this article, we consider practical coding and modulation schemes, and develop an analytical method to analyze effective throughput with finite block lengths and QPSK modulation. First, an analysis of the polarization effect between users in the RSMA system is conducted, with the impact of finite length coding taken into consideration. Thereafter, the effective throughput is proposed as the optimization objective. Effective throughput, which accurately reflects both system throughput and error probability, is a pivotal metric for designing IoT systems. Subsequently, to address this optimization problem, we propose an alternating optimization method that leverages successive convex approximation (SCA) and convex optimization in an iterative manner to find feasible solutions. Utilizing the optimized coding rates and precoding matrices, we design a polar-coded RSMA scheme for IoT application. Numerical results demonstrate that under QPSK modulation, RSMA can achieve up to a 33.08% gain in effective throughput compared with NOMA/SDMA, while polar-coded RSMA can reach up to 98.71% of the optimal performance target.

The forthcoming sixth-generation (6G) wireless system brings forth the need to meet increasingly stringent performance criteria, necessitating refinement and advanced design of channel coding and modulation techniques. Within this context, polar codes stand out as a competitive candidate due to their favorable attributes. This study delves into probabilistically-shaped polar-coded modulation (PS-PCM) for its potential to enhance spectral efficiency and error-correction performance, offering a pragmatic design geared toward standardization. Specifically, we present a coding chain for PS-PCM that ensures backward compatibility with legacy 5G polar codes. We also propose, for the first time, a channel-independent method for constructing PS-PCM that eliminates the need for on-the-fly computations, which is an imperative attribute for practical system implementation. The core of this code construction method is the utilization of a surrogate channel to allocate rates to each component code while considering the varying probability distributions and protection levels of label bits. The assigned component code rates can be seamlessly integrated into the modulation and coding scheme (MCS) table to facilitate rapid construction for both base stations and user equipment. Furthermore, we devise flexible rate matching methods to allow fine granularity adjustment of rates and blocklengths for PS-PCM. Simulation results validate the efficacy of our proposed approaches, which also provide insights and benchmarks for the capability of PS-PCM to meet requirements for enhanced spectral efficiency, higher reliability, and more flexibility.

In this paper, we propose an integrated polar coding scheme for both energy adaption and reliable transmission to enhance the existing results of joint energy and information transfer. More specifically, the encoding process is carried out by an energy adaptive encoder and a polar encoder. The energy adaptive encoder is implemented by a polar decoder to determine a subset of the bits, ensuring that the codewords passed through the polar encoder have an optimal distribution that achieves the energy-capacity function. The bits are allocated based on the conditional entropy derived from source and channel polarization. At the receiver, we slightly modify the decoder according to the prior distribution. Numerical simulations show that lk-block-length polar code achieves a gain of roughly or greater than 0.5 dB compared to the 100k-block-length low-density parity-check (LDPC) counterpart at a bit error rate (BER) of 10−3.

This paper conducts the first analysis of the bit error probabilityPbof polar codes under successive cancellation (SC) decoding, with a focus on information bits. We introduce the concept of component bit error probabilityPb(i), which corresponds to thei-th stage of SC decoding, andPbis computed as the sum of allPb(i) over the information set. To facilitate the evaluation ofPb(i) andPb, we propose an analytical approach that leverages the algebraic structure of a particular type of polar codes termed polar subcodes. This approach allows us to derive closed-form approximations for the upper bounds on bothPb(i) andPb. Our analysis applies to both systematic and non-systematic polar codes, where we further investigate the homogeneity and introduce the input-output weight distribution of polar subcodes to obtain more concise expressions for the proposed approximate upper bounds. Experimental results are also presented to validate the effectiveness of these approximations.

In this paper, we investigate the joint design and optimization of source-channel polar coding with 2m-ary transmission. A joint framework is proposed which includes a source polar code to compress the redundant source, followed by a set of component polar codes over a 2m-ary modulation to protect the source against errors and achieve increased spectral efficiency. We prove that our scheme suffices to achieve the theoretical limit of source-channel separation theorem. For finite-length cases, a joint decoder that exploits both the residual redundancy and channel characteristics is also derived to further reduce the error rate. Simulation results verify the effectiveness of the scheme.

Multilevel polar-coded modulation with multistage decoding is a capacity-achieving coded modulation scheme. In this paper, we propose a general formulation for analyzing the performance of multilevel polar-coded modulation and then derive the error probability upper bounds. The analysis explicitly reveals the effect of the modulation scheme on the performance of component polar codes. Based on the derived upper bounds, we also propose two construction methods for multilevel polar-coded modulation. Compared with conventional methods, such as density evolution and Gaussian approximation which involve complicated recursive calculations, the proposed methods have a linear computational complexity. Simulation results also show that the proposed construction methods can achieve comparable performance to existing methods under SC decoding, and even better performance under SC list decoding.

A rational leader selection strategy can enhance a swarm to manage the convergence and diversity during the entire search process. In this article, a novel adaptive multiobjective particle swarm optimization (MOPSO) is proposed on the basis of an evolutionary state estimation mechanism, which is used to detect the evolutionary environment whether in exploitation or exploration state. During the search process, different types of leaders, such as a convergence global best solution (c-gBest) and several diversity global best solutions (d-gBests), are to be selected from the external archive for particles under different evolutionary environments. The c-gBest is selected for improving the convergence when the swarm is in an exploitation state, while the d-gBests are chosen for enhancing the diversity in an exploration state. Furthermore, a modified archive maintenance strategy based on some predefined reference points is adopted to maximize the diversity of the Pareto solutions in the external archive. The experimental results demonstrate that the proposed algorithm performs significantly better than the several state-of-the-art multiobjective PSO algorithms and multiobjective evolutionary algorithms on 31 benchmark functions in terms of convergence and diversity of those obtained approximate Pareto fronts.

Code construction is a critical issue for polar coding. The expected construction method is of an accurate estimate of the reliability of bit-channels, but the current methods usually require high computational complexity. In this paper, we concern the input-output weight distribution of each bit-channel and derive its recursive calculation algorithm for systematic coding. The union bound and union-Bhattacharyya bound on the bit error probability are also derived to evaluate the reliability of bit-channels. Furthermore, by calculating the logarithmic form of the union-Bhattacharyya bound, we also propose two novel construction methods named the union-Bhattacharyya bound weight of the bit error probability (UBWB) and the simplified UBWB (SUBWB). Numerical results show that the proposed UBWB/SUBWB construction methods can achieve comparable performance to current methods under successive cancellation (SC) decoding and obtain obvious performance gain under SC list (SCL) decoding.

A many-objective problems (MaOP) refer to the optimization problem involving more than three objectives. Particle swarm optimization (PSO) is one of the potential heuristic methods suited for solving MaOPs. The personal best selection strategy, the global best selection strategy, and the archive maintenance strategy are the three key components in the design of a Many-Objective Particle Swarm Optimization (MaOPSO). The personal best and global best selection strategies determine the direction where particles will fly. The archive maintenance strategy has an important impact on convergence and diversity of its algorithm. In MaOPs, the high dimensionality in the objective space decreases the probability of a solution to be dominated by the other solutions in the population. Thus, it becomes more difficult for PSO to select the good leaders from so many non-dominated solutions. In this paper, a virtual Inverted Generational Distance indicator is proposed to evaluate the comprehensive quality of a solution in the external archive according to a constructed virtual Pareto front (vPF). Accordingly, a new indicator-based MaOPSO using vPF (MaOPSO/vPF) is developed to improve the convergence and diversity of the approximate Pareto front. Experimental results on the MaF test suites demonstrate that the proposed MaOPSO/vPF performs better than some selected competing Multi-objective Optimization Evolutionary Algorithms.