ALMA deep field in SSA22: Reconstructed [C II] luminosity function at z=6

The ADF22 line survey reported detections of two high-z line-emitting source candidates above $6\sigma$, both of which were shown to be spurious after follow-up observations. We investigate the detectability of far-infrared emitters in ALMA deep fields using mock observations by injecting artificial line-emitting sources into the visibility planes. We also discuss our investigation, conducted together with the ALMA operations team, of a possible technical problem in the original observations. Finally, we devise a method to estimate the [C ii] luminosity function (LF) at z similar to 6, including a full analysis of signal contamination and sample completeness. The comparison of pixel distributions between the real and mock datacubes does not show significant differences, confirming that the effect of non-Gaussian noise is negligible for the ADF22 datacube. Using 100 blank mock-mosaic datasets, we show 0.43 +/- 0.67 false detections per datacube with the previous source-finding method. We argue that the underestimation of the contamination rate in the previous work is caused by the smaller number of datacubes, using only four real ADF22 datacubes. We compare the results of clump-finding between the time division mode (TDM) and frequency division mode (FDM) correlator datacubes and confirm that the velocity widths of the clumps in the TDM case are up to three times wider than in the FDM case. Additional investigation into technical issues, specifically the "QA3 process," found no technical problems for both Cycle 2 and Cycle 4 data. Therefore, we confirm that false detections of high-SN ratios are unavoidable because clump-like structures already exist in large TDM datacubes. The LF estimation using our model shows that a correction for the number count is required, up to one order of magnitude, in the luminosity range of >= 5 x 10(8) L-circle dot. Our reconstruction method for the line LF can be applied to future blind line surveys.