  We present a study of shape, spectra and polarization properties of giant
pulses (GPs) from the Crab pulsar at the very high frequencies of 8.5 and 15.1
GHz. Studies at 15.1 GHz were performed for the first time. Observations were
conducted with the 100-m radio telescope in Effelsberg in Oct-Nov 2007 at the
frequencies of 8.5 and 15.1 GHz as part of an extensive campaign of
multi-station multi-frequency observations of the Crab pulsar. A selection of
the strongest pulses was recorded with a new data acquisition system, based on
a fast digital oscilloscope, providing nanosecond time resolution in two
polarizations in a bandwidth of about 500 MHz. We analyzed the pulse shapes,
polarisation and dynamic spectra of GPs as well as the cross-correlations
between their LHC and RHC signals. No events were detected outside main pulse
and interpulse windows. GP properties were found to be very different for GPs
emitted at longitudes of the main pulse and the interpulse. Cross-correlations
of the LHC and RHC signals show regular patterns in the frequency domain for
the main pulse, but these are missing for the interpulse GPs. We consider
consequences of application of the rotating vector model to explain the
apparent smooth variation in the position angle of linear polarization for main
pulse GPs.
  We also introduce a new scenario of GP generation as a direct consequence of
the polar cap discharge. We find further evidence for strong nano-shot
discharges in the magnetosphere of the Crab pulsar. The repetitive frequency
spectrum seen in GPs at the main pulse phase is interpreted as a diffraction
pattern of regular structures in the emission region. The interpulse GPs
however have a spectrum that resembles that of amplitude modulated noise.
Propagation effects may be the cause of the differences.
