## 1 Introduction  
Holographic data storage (HDS) has emerged as a promising technology for achieving high-density data storage with short read and write times [1]. In an HDS system, the information is stored as interference patterns or holograms created by the intersection of two laser beams. These page-oriented two-dimensional (2D) holograms can be stored and retrieved using various media, including photorefractive crystals and polymers. A key advantage of HDS over traditional data storage technologies is its potential for high storage density [2]. HDS can achieve storage densities of hundreds of terabytes per cubic inch owing to its ability to store multiple holograms in a single volume of storage medium. In addition, HDS has the potential for fast read and write times, making it suitable for high-performance computing applications. However, the development of HDS systems involves 2D inter-symbol interference (ISI), misalignments, and noise [3]. In addition, diffusion of the refractive index pattern over time can certainly have an impact on the detection of data in holographic data storage. When the refractive index pattern diffuses, it can cause the hologram to lose its sharpness and result in a decreased signal-to-noise ratio (SNR) in the detected signal. This can lead to errors in the data detection and potentially a loss of information. Researchers have attempted to overcome these problems in two ways: First, by enhancing the structure of the system. A pixel matching method was introduced by [PERSON] et al. [4]. [PERSON] et al. proposed an optimization of the aperture size [5]. [PERSON] et al. introduced aperture and medium defocusing [6]. The second category is the postprocessing and coding methods. Modulation methods for converting data bits into pixels in a hologram were introduced in some studies [7; 8; 9]. A combination of modulation and error-correcting codes was proposed to avoid isolated patterns and reduce the effect of 2D ISI [10; 11; 12; 13]. The isolated patterns are presented in Figure 1.  
In signal processing, an equalizer can be used to mitigate ISI. However, using a 2D equalizer directly to recover the original data results in a low bit error rate (BER) performance. Therefore, a generalized partial response (GPR) target was used with a 2D equalizer to estimate the channel parameters. GPR is a type of equalization technique that helps to reduce inter-symbol interference (ISI) caused by the channel response. These parameters are then used to detect the received signal in the original signal while maintaining a high BER performance. Initially, [PERSON] introduced the partial response maximum likelihood (PRML) method to balance and remove ISI in digital magnetic recording [14], which is a technology used for storing digital data on magnetic tape or other magnetic storage devices. In the PRML method, the received signal is converted into the desired target signal, which includes the known interference, by using an equalizer. Then, the output of the equalizer is supplied to the detection to decide the original data. [PERSON] and [PERSON] proposed a GPR target to improve the detection performance of the PRML [15]. Regularly, researchers have used the one-dimensional (1D) form for PR and GPR targets because calculating all Viterbi paths is very complicated when using the 2D form for PR and GPR targets. Therefore, many studies have used and modified PRML and GPR to design and improve 2D detection for storage systems [16; 17; 18; 19]. Recently, [PERSON] and [PERSON] proposed a serial target for bit-patterned media recording (BPMR) systems based on the 2D GPR target and the serial structure of detection [20; 21; 22]. Serial detection is based on the serial form of the 2D target. After estimating the 2D target, it was decomposed into a series of two 1D targets. In the serial detection structure, the first detector is used to estimate the values of six possible target symbols.