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Aims. We present a study of shapes, 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 are performed for the first time. We probe GP emission at high frequencies and examine its intrinsic spectral and polarization properties with high time and spectral resolution. The use of high radio frequencies also alleviates the effects of pulse broadening caused by interstellar scattering, which masks the intrinsic properties of GPs at low frequencies. Methods. 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 with a bandwidth of about 500 MHz. In total, 29 and 85 GPs at longitudes of the main pulse and interpulse were recorded at 8.5 and 15.1 GHz during 10 and 17 h of observing time respectively. We analyzed the pulse shapes, polarisation and dynamic spectra of GPs as well as the cross-correlations between their LHC and RHC signals. Results. No events were detected outside the main pulse and interpulse windows. The 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, which are missing for the interpulse GPs. We consider the consequences of applying 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. Conclusions. We find further evidence of 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.

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