The detection of six Fast Radio Bursts (FRBs) has recently been reported. FRBs are short duration ($sim$ 1 ms), highly dispersed radio pulses from astronomical sources. The physical interpretation for the FRBs remains unclear but is thought to involve highly compact objects at cosmological distance. It has been suggested that a fraction of FRBs could be physically associated with gamma-ray bursts (GRBs). Recent radio observations of GRBs have reported the detection of two highly dispersed short duration radio pulses using a 12 m radio telescope at 1.4 GHz. Motivated by this result, we have performed a systematic and sensitive search for FRBs associated with GRBs. We have observed five GRBs at 2.3 GHz using a 26 m radio telescope located at the Mount Pleasant Radio Observatory, Hobart. The radio telescope was automated to rapidly respond to Gamma-ray Coordination Network notifications from the Swift satellite and slew to the GRB position within $sim$ 140 s. The data were searched for pulses up to 5000 pc $rm cm^{-3}$ in dispersion measure and pulse widths ranging from 640 $rm mu$s to 25.60 ms. We did not detect any events $rm geq 6 sigma$. An in-depth statistical analysis of our data shows that events detected above $rm 5 sigma$ are consistent with thermal noise fluctuations only. A joint analysis of our data with previous experiments shows that previously claimed detections of FRBs from GRBs are unlikely to be astrophysical. Our results are in line with the lack of consistency noted between the recently presented FRB event rates and GRB event rates.
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