We present the survey design, data reduction, construction of images, and source catalog of the Atacama Large Millimeter/submillimeter Array (ALMA) twenty-six arcmin^2 survey of GOODS-S at one-millimeter (ASAGAO). ASAGAO is a deep (1sigma ~ 61 uJy/beam for a 250 klambda-tapered map with a synthesized beam size of 0.51 x 0.45) and wide area (26 arcmin^2) survey on a contiguous field at 1.2 mm. By combining with ALMA archival data in the GOODS-South field, we obtained a deeper map in the same region (1sigma ~ 30 uJy/beam for a deep region with a 250 klambda-taper, and a synthesized beam size of 0.59 x 0.53), providing the largest sample of sources (25 sources at >=5.0sigma, 45 sources at >=4.5sigma) among ALMA blank-field surveys to date. The number counts shows that 52(+11 -8)% of the extragalactic background light at 1.2 mm is resolved into discrete sources at S1.2m > 135 uJy. We create infrared (IR) luminosity functions (LFs) in the redshift range of z = 1-3 from the ASAGAO sources with KS-band counterparts, and constrain the faintest luminosity of the LF at 2.0 < z < 3.0. The LFs are consistent with previous results based on other ALMA and SCUBA-2 observations, which suggest a positive luminosity evolution and negative density evolution with increasing redshift. We find that obscured star-formation of sources with IR luminosities of log(L(IR)/Lsun)} ~> 11.8 account for ~~60%-90% of the z ~ 2 cosmic star-formation rate density.
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