We report a limit on the ultra-high-energy neutrino flux based on a non-detection of radio pulses from neutrino-initiated particle cascades in the Moon, in observations with the Parkes radio telescope undertaken as part of the LUNASKA project. Due to the improved sensitivity of these observations, which had an effective duration of 127 hours and a frequency range of 1.2-1.5 GHz, this limit extends to lower neutrino energies than those from previous lunar radio experiments, with a detection threshold below 10^20 eV. The calculation of our limit allows for the possibility of lunar-origin pulses being misidentified as local radio interference, and includes the effect of small-scale lunar surface roughness. The targeting strategy of the observations also allows us to place a directional limit on the neutrino flux from the nearby radio galaxy Centaurus A.
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