We present a CARMA 1.3 mm continuum survey toward 9 Class 0 protostars in the Perseus molecular cloud at $sim$0.3$^{primeprime}$ (70 AU) resolution. This study approximately doubles the number of Class 0 protostars observed with spatial resolutions $<$ 100 AU at millimeter wavelengths, enabling the presence of protostellar disks and proto-binary systems to be probed. We detect flattened structures with radii $>$ 100 AU around 2 sources (L1448 IRS2 and Per-emb-14) and these sources may be strong disk candidates. Marginally-resolved structures with position angles within 30$^{circ}$ of perpendicular to the outflow are found toward 3 protostars (L1448 IRS3C, IRAS 03282+3035, and L1448C) and are considered disk candidates. Two others (L1448 IRS3B and IRAS 03292+3039) have resolved structure, possibly indicative of massive inner envelopes or disks; L1448 IRS3B also has a companion separated by 0.9$^{primeprime}$ ($sim$210 AU). IC348-MMS does not have well-resolved structure and the candidate first hydrostatic core L1451-MMS is marginally resolved on 1$^{primeprime}$ scales. The strong disk candidate sources were followed-up with C$^{18}$O ($J=2rightarrow1$) observations, detecting velocity gradients consistent with rotation, but it is unclear if the rotation is Keplerian. We compare the observed visibility amplitudes to radiative transfer models, finding that visibility amplitude ratios suggest a compact component (possibly a disk) is necessary for 5 of 9 Class 0 sources; envelopes alone may explain the other 4 systems. We conclude that there is evidence for the formation of large disks in the Class 0 phase with a range of radii and masses dependent upon their initial formation conditions.
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