## 1 Introduction  
In the field of data storage, conventional magnetic recording systems have an area density limit of \(\sim\)1 Tb/in\({}^{2}\) because of the superparamagnetic effect [1]. Therefore, bit-patterned media recording (BPMR) has been proposed as a promising candidate to solve this problem. In BPMR, one bit is stored in a single-domain island surrounded by a nonmagnetic region, and the areal density (AD) can reach up to 4 Tb/in\({}^{2}\)[2]. Moreover, BPMR technology can decrease the nonlinear transition shift, offer easier tracking, and exhibit good thermal stability [2; 3]. For increasing the storage capacity, the distance between magnetic islands must be closer, which produces intersymbol interference (ISI) and intertrack interference (ITI) from the down-track (horizontal) and cross-track (vertical) directions, respectively. These types of interference are referred to as two-dimensional (2D) interference in BPMR systems.  
Many methods are used to reduce 2D interference. We can use modulation methods like those of [PERSON] [4], with an error-correcting 5/6 code to avoid fatal interference as much as possible and gain error correction; of [PERSON] [5], with a combination of a rate-3/4 modulation code and ITI subtraction to reduce ITI; and of [PERSON] [6], with a rate-5/6 constructive ITI code to handle ITI. We can use detection in combination with a partial response (PR) target and an equalizer. To reduce ITI, [PERSON] [7] proposed a modified trellis for BPMR. In [8], [PERSON] suggested a 2D equalizer with a PR target and optimized equalizer coefficients to eliminate ISI and ITI. Based on [PERSON] [7; 8], a hybrid 2D equalizer was proposed by [PERSON] [9] to improve channel estimation when ITI is known. Additionally,to mitigate 2D interference, [PERSON] [10] proposed a 2D soft-output Viterbi algorithm (2D SOVA), which is also known as a parallel detection model, for holographic data storage, then developed an iterative 2D SOVA for bit-patterned media [11]. [PERSON], based on the parallel detection model, proposed a multipath ISI structure that fits the staggered BPMR structure [12]. In addition to the idea of parallel detection, recently the idea of serial detection appeared [13]. In serial detection, the 2D interference is separated into two serial interference components. This helps reduce the complexity of 2D Viterbi detection of the BPMR channel with 2D interference. However, in serial detection, the inner detection still has an output that is referred to as a hard output. This is a disadvantage of serial detection compared to parallel detection, which has a soft output in inner detection.  
In this study, we propose two methods to create a soft output for the inner detector. The first method is to use an equalizer to create a soft output. The second method is based on the Viterbi algorithm combined with channel interference to create a soft output. This method is highly effective, because it can reduce errors compared to a six-level signal and preserve the interference information of the signal. Moreover, when the channel experiences track mis-registration (TMR), which degrades system performance [14; 15], the previous PR target is no longer effective in overcoming TMR. Therefore, we have proposed a new form of PR target for the 2D BPMR channel with TMR. Finally, we investigated the ability of serial detection in the presence of media noise.  
The rest of this paper is organized as follows. In Section 2, we explain our algorithm and show how we design the soft output for serial detection. Section 3 briefly presents the BPMR channel model and illustrates the simulation results. Finally, the conclusion is drawn in Section 4.