## 3 RBER Estimation  
The RBER is estimated by comparing the true-value sequence inserted into specific positions of each frame with the one read from the flash to get an estimated FER. To ensure the accuracy of the estimated RBER, averaging the estimated FERs of multiple sample frames is required. The accuracy depends on two factors, the length of the true-value sequence \(L\) and the number of sample frames \(N\). With the interleaving module to balance the errors of frames from different pages, there will be no huge difference in FERs obtained from the frames. Otherwise, it may result in inaccurate estimation in HSD NAND flash. We estimated RBER with the mean of \(N=16\) sample frames because the division by 16 in estimation can be implemented with a right shifter. Moreover, it is a short time to wait and collect 16 frames in the HSD NAND flash, especially when the parallel structure of encoding and decoding is used, so the RBER of the flash can be reflected in time. The larger \(N\) can certainly lead to more accurate estimation, but it will take longer. The length \(L\) of the true-value sequence should also be selected carefully because a too-long sequence will cause too much rate penalty, whereas a too-short one will cause inaccurate estimation. Figure 7 shows how the lengths of the true-value sequence affect the mean square error (MSE) between \(\overline{BER_{est}}\), the estimated BER, and the actual RBER when tested with a (10,080, 8400) QC-LDPC from [22]. The MSE is about \(1.7\times 10^{-5}\) at the length of 200/8400, about 5 times larger than \(3.4\times 10^{-6}\) at 300/8400, and then stays at the same order of magnitude down to \(2.9\times 10^{-6}\) at 400/8400. Namely, the MSE is decreasing with the increase of the length, but the trend is gradually slowing down when the length exceeds 300/8400. To balance the rate penalty and the accuracy of estimation, the length is suggested to be less than 3.5% of the total length of the frame. We set \(L=256\) in our test for the convenience of hardware implementation. Figure 8 illustrate two cases of estimation at RBER = 0.01 and 0.1. In the HSD flash using 1/2 gray coding modulation, the estimation is very inaccurate when directly comparing data without interleaving. In this case, only 55.49% and 36.99% of the estimated values fall in [0.009, 0.011] and [0.09, 0.11], respectively, the \(\pm 10\%\) range of the actual RBER. The interleaving module greatly improves the accuracy of the estimation for such flashes, making 85.37% and 99.55% of the estimated values fall within [0.009, 0.011] and [0.09, 0.11], respectively, and significantly narrowing the distribution interval. For the flash using balanced gray coding modulation, where about 83.4% and 97.89% of the estimated values are located within [0.009, 0.011] and [0.09, 0.11], respectively, interleaving only slightly increases the accuracy and narrows the distribution interval. Nevertheless, in general, interleaving can improve the accuracy of RBER estimation in HSD NAND flash using either modulation. Figure 9 shows that with the interleaving, the method can make \(\overline{BER_{est}}\) very close to the actual RBER, and they are highly linearly correlated though there exist some small fluctuations. Therefore, it is credible to estimate RBER in NAND flash with the true-value insertion method equipped with interleaving, and the good linearity can make the \(\overline{BER_{est}}\) sensitive to RBER change. Hence, it is suitable to measure and reflect RBER in HSD NAND flash in time. \(\overline{BER_{est}}\) can help select decoding statuses. Usually, the soft-decision decoding is selected after the default reading level with hard-decision decoding and retry level with hard-decision decoding are failed, which wastes time on trial and error. Based on the current BER estimated, the decoding status can be directly selected so that the controller does not have to try through all decoding statuses, which thereby increase the reading speed for NAND flash.