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We describe an experiment using the Parkes radio telescope in the 1.2–1.5 GHz frequency range as part of the LUNASKA project, to search for nanosecond-scale pulses from particle cascades in the Moon, which may be triggered by ultra-high-energy astroparticles. Through the combination of a highly sensitive multi-beam radio receiver, a purpose-built backend and sophisticated signal-processing techniques, we achieve sensitivity to radio pulses with a threshold electric field strength of 0.0053 μ V/m/MHz, lower than previous experiments by a factor of three. We observe no pulses in excess of this threshold in observations with an effective duration of 127 h. The techniques we employ, including compensating for the phase, dispersion and spectrum of the expected pulse, are relevant for future lunar radio experiments.