## 2 BPMR Channel Model  
This article presents a staggered island BPMR medium using a three-reader one-track detection scheme, as illustrated in Figure 1. The readback signal from the desired track experiences ITI from both upper and lower adjacent tracks, while these adjacent tracks similarly experience ITI from the desired track. Thus, we expect that some information from the desired track is embedded within the readback signals of the adjacent tracks, creating a form of mutual information. A key contribution is to utilize this mutual information to improve the quality of the estimated user bits, thereby enhancing overall system performance. Figure 2 depicts the cross-sectional view of the head-medium geometry in the staggered BPMR system used in this study. The soft-magnetic underlayer (SUL) thickness was 300 nanometers (nm), the non-magnetic interlayer (IL) thickness was 15 nm, the recording layer (RL) thickness was 10 nm, and the head-medium spacing (HMS) was 5 nm, which was the distance between the air-bearing surface (ABS) of the read/write head and the top surface of the RL. To model the staggered BPMR medium, as shown in Figure 3, we arranged a series of circular magnetic islands, each with a diameter of 10 nm, in a staggered pattern on a non-magnetic surface, represented by the white region. The gray and black islands indicate  
Figure 1: Staggered bit-patterned recording medium and three-reader array at an AD of 3.0 Tb/in\({}^{2}\). Figure 2: Cross section of head-medium geometry for the staggered recording layer BPMR medium and an array head with three readers. magnetization directions of \"down\" (\(-1\)) and \"up\" (\(+1\)), respectively. Figure 3 shows the center reader among three readers packed in an array reader, with its read head sensitivity focused on reading data from the desired track. The read head sensitivity extends to adjacent tracks, which causes the ITI effect. The AD was determined by the bit period, \(T_{x}\), and the track pitch, \(T_{z}\). For example, the AD equals 3.0 Tb/in\({}^{2}\) when \(T_{x}=T_{z}=14.5\) nm. Additionally, bit island size and position fluctuations were incorporated as sources of media noise to provide a more realistic BPMR channel model. Specifically, the size fluctuation was defined as  
\[\text{size fluctuation}=\frac{\sigma_{d}}{d}, \tag{1}\]  
where \(d\) is the ideal diameter of the bit island, and \(\sigma_{d}\) is the standard deviation of the bit island diameter (a Gaussian distribution). Moreover, the position fluctuation was defined as  
\[\text{position fluctuation}=\frac{\sigma_{T_{x}}}{T_{x}}, \tag{2}\]  
where \(T_{x}\) is the bit period, and \(\sigma_{T_{x}}\) is the standard deviation of the distance between the centers of the bit islands (a Gaussian distribution). Note that the size and position fluctuations in this study are independent and identically distributed (_i.i.d_) random variables. This paper considers the staggered medium for three-reader one-track detection depicted in Figure 1 in conjunction with the channel model shown in Figure 4. A 4096-bit user sequence, \(u_{k,l}\in\{\pm 1\}\), where the bits are defined as _i.i.d_ random variables, commonly used in various studies of magnetic recording channels, was recorded onto the medium. All user bits of the considered tracks are generated randomly before being sent to the writing process. A perfect writing process was assumed, in which the magnetization of the bit islands is always the same as the intended bit sequence. During a reading process, an array of three read heads simultaneously detects these three data tracks (or uses a single read head to sequentially detect these tracks with the help of buffer memory), as illustrated in Figure 1. As for our read head arrangement, the middle reader (H2) was positioned at the center of track \(l\), referred to as the desired track or middle track. The upper reader (H1) was placed between tracks (\(l-1\)) and \(l\), while the lower reader (H3) was placed between tracks \(l\) and (\(l+1\)).