## 1 Introduction  
Currently, the rapid development in data science has led to an ever-increasing amount of data. Therefore, the demand for data storage has accordingly increased. Storage technologies are of two types, namely solid-state storage and magnetic storage. In solid-state storage, the data are stored as electrons in floating gates [1], whereas in magnetic storage, the data are magnetized and stored in a magnetized medium [2]. The choice between solid-state storage and magnetic storage involves a trade-off between reading speed and cost per bit [3]. The initial recording medium was a magnetic material with many randomly oriented magnetic grains. Therefore, it is limited to an areal density (AD) of approximately 1 Tb/in\({}^{2}\) by superparamagnetic phenomena [4]. To overcome this problem, several new technologies have been proposed, including heat-assisted magnetic recording (HAMR) [5], two-dimensional (2D) magnetic recording (TDMR) [2], and bit-patterned media recording (BPMR) [6]. Among these, BPMR is a promising candidate for future technology. For BPMR, the recording media structure is based on magnetic islands surrounded by nonmagnetic zones. To increase the AD, we need to reduce the area of the nonmagnetic region, thus making the magnetic islands closer. However, with closer distances, BPMR experiences more two-dimensional (2D) interference. The 2D interference consists of intersymbol interference (ISI) and intertrack interference (ITI) upon reducing the distance according to the down-track and cross-track directions, respectively. Furthermore, BPMR suffers from track misregistration (TMR) and media noise [7, 8]. Many signal-processing algorithms have been proposed to address 2D interference. Typically, there are two common approaches, namely symbol detection methods and error-correcting or/and modulation coding methods. The key concept behind the modulation codes is that the symbols must not be surrounded by the opposite polarity symbols to prevent 2D interference. However, these symbols can have room for error correction if the minimum distance of the code is greater than one. To reduce the ITI, [PERSON] and [PERSON] [9] designed a rate 3/4 modulation code that subtracts the ITI. Based on the staggered BPMR, [PERSON] and [PERSON] proposed an error-correcting 5/6 modulation code to reduce the isolated bit and correct the errors [10]. [PERSON] and [PERSON] [11] introduced a modulation code and its multilayer perceptron decoding for BPMR to reduce and correct the errors by the ITI. For symbol detection, maximum likelihood (ML) detection and maximum a priori (MAP) detection are commonly used for white noise and colored noise channels, respectively. The Viterbi algorithm (VA) and Bahl-Cocke-Jelinek-Raviv (BCJR) algorithms are representative of ML and MAP detections, respectively. In principle, the Viterbi and BCJR algorithms are designed to remove 1D interference, such as the ISI. However, in the BPMR, the signal suffers from 2D interference; therefore, we need to modify Viterbi and BCJR to be suitable for 2D interference. [PERSON] and [PERSON] [12] proposed a model that includes a 2D equalizer and a generalized partial response (GPR) target to estimate the interference. Consequently, the interference value is applied for detection to efficiently remove the interference in the signal. In addition, [PERSON] et al. introduced a modified Viterbi algorithm (MVA) to supply ITI information for the VA [13]. Thus, MVA can reduce the ITI and improve the BER performance of BPMR systems. In addition to the improvements in [12; 13], [PERSON] and [PERSON] combined GPR and MVA to enhance the channel estimation when the ITI characteristic is known, and the method is referred to as a hybrid 2D equalizer [14]. Further, to mitigate 2D interference, [PERSON] and [PERSON] [15] suggested a 2D soft-output VA (2D SOVA). In 2D SOVA, the received signal is detected in parallel by two 1D Viterbi detectors applied in the horizontal and vertical directions, respectively.