Context. Observations of neutral hydrogen can provide a wealth of information about the kinematics of galaxies. To learn more about the large-scale structures and accretion processes, the extended environment of galaxies have to be observed. Numerical simulations predict a cosmic web of extended structures and gaseous filaments.Aims. To observe the direct vicinity of galaxies, column densities have to be achieved that probe the regime of Lyman limit systems. Typically, HI observations are limited to a brightness sensitivity of N-HI similar to 10(19) cm(-2), but this has to be improved by similar to 2 orders of magnitude.Methods. With the Westerbork Synthesis Radio Telescope (WSRT), we mapped the galaxy filament connecting the Virgo Cluster with the Local Group. About 1500 square degrees on the sky was surveyed with Nyquist sampled pointings. By using the WSRT antennas as single-dish telescopes instead of the more conventional interferometer, we were very sensitive to extended emission. The survey consists of a total of 22 000 pointings, and each pointing was observed for two minutes with 14 antennas.Results. We reached a flux sensitivity of 16 mJy beam(-1) over 16 km s(-1), corresponding to a brightness sensitivity of N-HI similar to 3.5 x 10(16) cm(-2) for sources that fill the beam. At a typical distance of ten Mpc probed by this survey, the beam extent corresponds to about 145 kpc on linear scale. Although the processed data cubes are affected by confusion owing to the very large beam size, we can identify most of the galaxies that have been observed in HIPASS. Furthermore we made 20 new candidate detections of neutral hydrogen. Several of the candidate detections can be linked to an optical counterpart. The majority of the features, however, do not show any signs of stellar emission. Their origin is investigated further with accompanying HI surveys, which will be published in follow-up papers.