We present measurements of Polar Mesospheric Summer Echoes, PMSE, at the very short Bragg wavelength of 16 cm which occur infrequently. The measurements were carried out with the EISCAT 930 MHz UHF radar. Part of the data were taken under the influence of HF heating. A comprehensive comparison of the measurements with the theory of turbulence including enhancement of the Schmidt number is carried out, but other theories are also considered. Estimates of the energy dissipation rates inferred from the signal spectral widths are used to calculate Batchelor scale lengths as a function of the Schmidt number. Hill's multipolar diffusion model is used to calculate the diffusion rates (which depend on the charge number of ice particles of nanometer size radius) leading to estimates of the Schmidt numbers. It is argued that the use of the slow diffusion coefficient in the calculations should give conservative estimates, i.e., lower bounds, of the necessary ice charge numbers to explain the measurements, the condition for enhanced scattering being that the Bragg scale of the radar should be greater than the Batchelor scale. Based on information from other experiments published in the literature, e.g., the charge number of mesospheric ice particles, and theoretical considerations, the main conclusion of this investigation is that none of the most prominent theories can explain satisfactorily the measurements unless charge numbers in excess of 10 electrons for the measured wide spectra and at least in excess of 100s of electrons for the measured narrow spectra are assumed. The HF heating effect on the mesosphere appears to have a similar influence on PMSE at a Bragg wavelength of 16 cm as it has at greater wavelengths, namely a weakening of PMSE during the heater-on periods and a recovery during the heater off-periods. This behaviour is as expected given the rapidity and large amount of electron heating that the HF wave produces with the consequent rapid and large enhancement of the electron diffusion rate during heater-on periods regardless of the electron diffusion model or reference to any particular model of PMSE.