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Previously, we have shown that the transition position shift in a perpendicular recording medium with a recording layer and a soft underlayer (SUL) is one of the major noise sources in high-density recording when abrupt single pole head field is applied. In this paper, we have considered a head rise time calculated from micromagnetic calculation for three different damping constants of pole tip. Also, in order to reduce the transition position shift, a modified medium with a synthetic antiferromagnetically coupled (SAFC) SUL was investigated. The SAFC-SUL was assumed to the 20-nm-thick top SUL with Ms of 800 emu/cc and the 80-nm-thick bottom-SUL with magnetization saturation (Ms) of 1200 emu/cc separated by a 0.6 nm Ru layer. The media with SAFC-SUL showed lower noise and higher signal-to-noise ration than the media with conventional SUL. The lower noise in the SAFC-SUL seems to be associated with suppressed vortex formation in the SUL by synthetic antiferromagnetic coupling in the SUL.