# A Simple Neural-Network-Based Decoder for Short Binary Linear Block Codes
## 1 Introduction  
The study of the application on neural networks to error correction codes has been ongoing since the late 1980s [1; 2; 3; 4; 5; 6; 7; 8; 9]. However, at that time, hardware technology needed to be more capable of efficiently training and implementing sizable neural network (NN) models, which limited the number of network layers and performance. The training results thus often did not reach the desired bit error rate. In recent years, advances in hardware technology have made it possible to train and implement more extensive and complex NN models, leading to significant progress in this field. The use of deep learning for error correction codes has become more popular in recent years due to the rise of deep learning (DL) and the availability of hardware acceleration. The NNs for decoding are typically divided into two main categories: binary classification and noise elimination architectures. The binary classification architecture is designed to classify the input data into two categories, such as \"correct\" and \"incorrect,\" while the noise elimination architecture aims to remove noise from the input data to improve the decoding accuracy. Both approaches have effectively improved the performance of error correction codes and are an active area of research. Several soft decision decoding (SDD) methods are commonly used in error correction codes. These include the belief propagation (BP) algorithm [10], which uses an iterative method to eliminate noise in the input signal, and the Chase-II algorithm [11], which uses enumeration to select candidate codewords obtained by the hard decision decoder (HDD). Another popular method is the ordered statistics decoding (OSD) method [12],which reconstructs the generator matrix of the code based on the bit reliability to produce the possible candidate message. The candidate codewords are finally selected from the perspective of the maximum likelihood. Traditional SDD decoding methods often require significant time and computing resources, as they rely on iterative, enumerative, and sorting processes. In contrast to conventional SDD methods, the DL-based methods require only matrix multiplication to obtain a solution as the NN or deep neural network (DNN) is trained. This method greatly simplifies the calculation process, and hardware-accelerated technology can be used to perform parallelized calculations, significantly reducing the time required for error correction. In addition, the DL-based methods can learn and adapt to various codes. These advantages make the DL-based method an attractive alternative to traditional methods. The DNN-based decoder presented in [13] eliminated the problem of overfitting to the training codeword set by using the syndrome and channel reliabilities as the input of the network. This framework makes the NN focus on the noise estimation. In [14], a novel DNN-based denoiser was presented, which directly learns the mapping from a noisy codeword to its corresponding denoised one. Results showed that the effectiveness of the denoiser is significant. Authors in [5] proposed the DNN-based belief propagation flip (BPF) decoder for polar codes. The idea is to deploy a DNN to decide which bits to flip. In [7], the convolutional neural network (CNN) model is employed for decoding polar codes to reduce the delay and efficiency. [PERSON] et al. [8] introduced a simplified metric derived from the path metric domain and designed a custom-tailored DNN to enhance its efficiency when the successive cancelation list (SCL) decoder is adopted. Simulation results indicated that such a metric incurs almost no performance loss but with lower computational complexity. In [6], a model-driven DL decoder for irregular binary low-density parity-check (LDPC) codes was invented by the alternating direction method of multipliers (ADMM) technique.