The Thousand-Pulsar-Array programme on MeerKAT III: Giant pulse characteristics of PSR J0540$-$6919

PSR J0540$-$6919 is the second-most energetic radio pulsar known and resides in the Large Magellanic Cloud. Like the Crab pulsar it is observed to emit giant radio pulses (GPs). We used the newly-commissioned PTUSE instrument on the MeerKAT radio telescope to search for GPs across three observations. In a total integration time of 5.7 hrs we detected 865 pulses above our 7$\sigma$ threshold. With full polarisation information for a subset of the data, we estimated the Faraday rotation measure, $\rm{RM}=-245.8 \pm 1.0$ rad m$^{-2}$ toward the pulsar. The brightest of these pulses is $\sim$ 60% linearly polarised but the pulse-to-pulse variability in the polarisation fraction is significant. We find that the cumulative GP flux distribution follows a power law distribution with index $-2.75 \pm 0.02$. Although the detected GPs make up only $\sim$ 10% of the mean flux, their average pulse shape is indistinguishable from the integrated pulse profile, and we postulate that there is no underlying emission. The pulses are scattered at L-band frequencies with the brightest pulse exhibiting a scattering time-scale of $\tau = 0.92 \pm 0.02$ ms at 1.2 GHz. We find several of the giants display very narrow-band "flux knots" similar to those seen in many Fast Radio Bursts, which we assert cannot be due to scintillation or plasma lensing. The GP time-of-arrival distribution is found to be Poissonian on all but the shortest time-scales where we find four GPs in six rotations, which if GPs are statistically independent is expected to occur in only 1 of 7000 observations equivalent to our data.