We present results of a ^{12}CO J = 3-2 survey of 125 nearby galaxies obtained with the 10-m Heinrich-Hertz-Telescope, with the aim to characterize the properties of warm and dense molecular gas in a large variety of environments. With an angular resolution of 22, ^{12}CO 3-2 emission was detected in 114 targets. Based on 61 galaxies observed with equal beam sizes the ^{12}CO 3-2/1-0 integrated line intensity ratio R_{31} is found to vary from 0.2 to 1.9, with an average value of 0.81. No correlations are found for R_{31} to Hubble type and far infrared luminosity. Possible indications for a correlation with inclination angle and the 60mum/100mum color temperature of the dust are not significant. Higher R_{31} ratios than in ``normal galaxies, hinting at enhanced molecular excitation, may be found in galaxies hosting active galactic nuclei. Even higher average values are determined for galaxies with bars or starbursts, the latter being identified by the ratio of infrared luminosity versus isophotal area, log[(L_{FIR}/L_{SUN})/(D_{25}/kpc)^2)] > 7.25. (U)LIRGs are found to have the highest averaged R_{31} value. This may be a consequence of particularly vigorous star formation activity, triggered by galaxy interaction and merger events. The nuclear CO luminosities are slightly sublinearly correlated with the global FIR luminosity in both the ^{12}CO J = 3-2 and the 1-0 lines. The slope of the log-log plots rises with compactness of the respective galaxy subsample, indicating a higher average density and a larger fraction of thermalized gas in distant luminous galaxies. While linear or sublinear correlations for the ^{12}CO J = 3-2 line can be explained, if the bulk of the observed J = 3-2 emission originates from molecular gas with densities below the critical one, the case of the ^{12}CO J = 1-0 line with its small critical density remains a puzzle.
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