In this work, we introduce β-expansion, a notion borrowed from number theory, as a theoretical framework to study fast construction of polar codes based on a recursive structure of universal partial order (UPO) and polarization weight (PW) algorithm. We show that polar codes can be recursively constructed from UPO by continuously solving several polynomial equations at each recursive step. From these polynomial equations, we can extract an interval for β, such that ranking the synthetic channels through a closed-form βexpansion preserves the property of nested frozen sets, which is a desired feature for low-complex construction. In an example of AWGN channels, we show that this interval for β converges to a constant close to 1.1892 ≈ 2 1/4 when the code block-length trends to infinity. Both asymptotic analysis and simulation results validate our theoretical claims.

In this paper, we investigate an artificialintelligence (AI) driven approach to design error correction codes (ECC). Classic error-correction code design based upon coding-theoretic principles typically strives to optimize some performance-related code property such as minimum Hamming distance, decoding threshold, or subchannel reliability ordering. In contrast, AI-driven approaches, such as reinforcement learning (RL) and genetic algorithms, rely primarily on optimization methods to learn the parameters of an optimal code within a certain code family. We employ a constructor-evaluator framework, in which the code constructor can be realized by various AI algorithms and the code evaluator provides code performance metric measurements. The code constructor keeps improving the code construction to maximize code performance that is evaluated by the code evaluator. As examples, we focus on RL and genetic algorithms to construct linear block codes and polar codes. The results show that comparable code performance can be achieved with respect to the existing codes. It is noteworthy that our method can provide superior performances to classic constructions in certain cases (e.g., list decoding for polar codes). Code PerformanceCoding Theory Code Construction AI Techniques Fig. 1: Error correction code design logic improve its code performance. Equivalently, given a target error rate, we optimize code design to maximize the achievable code rate, i.e. to approach the channel capacity. A. Code design based on coding theoryClassical code construction design is built upon coding theory, in which code performance is analytically derived in terms of various types of code properties. To tune these properties is to control the code performance so that code design problems are translated into code property optimization problems.Hamming distance is an important code property for linear block codes of all lengths. For short codes, it is the dominant factor in performance, when maximum-likelihood (ML) decoding is feasible. For long codes, it is also important for performance in the high signal-to-noise ratio (SNR) regime. A linear block code can be defined by a generator matrix G or the corresponding parity check matrix H over finite fields. Directed by the knowledge of finite field algebra, the distance profile of linear block codes can be optimized, and in particular, the minimum distance. Examples include Hamming codes, Golay codes, Reed-Muller (RM) codes, quadratic residue (QR) codes, Bose-Chaudhuri-Hocquenghem (BCH) codes, Reed-Solomon (RS) codes, etc.Similar to the Hamming distance profile, free distance, another code property, is targeted for convolutional codes. Convolutional codes [2] are characterized by code rate and the memory order of the encoder. By increasing the memory order and selecting proper polynomials, larger free distance can be obtained at the expense of encoding and decoding

Integrating artificial intelligence (AI) into wireless networks has drawn significant interest in both industry and academia. A common solution is to replace partial or even all modules in the conventional systems, which is often lack of efficiency and robustness due to their ignoring of expert knowledge. In this paper, we take deep reinforcement learning (DRL) based scheduling as an example to investigate how expert knowledge can help with AI module in cellular networks. A simulation platform, which has considered link adaption, feedback and other practical mechanisms, is developed to facilitate the investigation. Besides the traditional way, which is learning directly from the environment, for training DRL agent, we propose two novel methods, i.e., learning from a dual AI module and learning from the expert solution. The results show that, for the considering scheduling problem, DRL training procedure can be improved on both performance and convergence speed by involving the expert knowledge. Hence, instead of replacing conventional scheduling module in the system, adding a newly introduced AI module, which is capable to interact with the conventional module and provide more flexibility, is a more feasible solution.

In this paper, we model nested polar code construction as a Markov decision process (MDP), and tackle it with advanced reinforcement learning (RL) techniques. First, an MDP environment with state, action, and reward is defined in the context of polar coding. Specifically, a state represents the construction of an (N, K) polar code, an action specifies its reduction to an (N, K − 1) subcode, and reward is the decoding performance. A neural network architecture consisting of both policy and value networks is proposed to generate actions based on the observed states, aiming at maximizing the overall rewards. A loss function is defined to trade off between exploitation and exploration. To further improve learning efficiency and quality, an "integrated learning" paradigm is proposed. It first employs a genetic algorithm to generate a population of (sub-)optimal polar codes for each (N, K), and then uses them as prior knowledge to refine the policy in RL. Such a paradigm is shown to accelerate the training process, and converge at better performances. Simulation results show that the proposed learning-based polar constructions achieve comparable, or even better, performances than the state of the art under successive cancellation list (SCL) decoders. Last but not least, this is achieved without exploiting any expert knowledge from polar coding theory in the learning algorithms.

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