  LOPES, the LOFAR prototype station, was an antenna array for cosmic-ray air
showers operating from 2003 - 2013 within the KASCADE-Grande experiment.
Meanwhile, the analysis is finished and the data of air-shower events measured
by LOPES are available with open access in the KASCADE Cosmic Ray Data Center
(KCDC). This article intends to provide a summary of the achievements, results,
and lessons learned from LOPES. By digital, interferometric beamforming the
detection of air showers became possible in the radio-loud environment of the
Karlsruhe Institute of Technology (KIT). As a prototype experiment, LOPES
tested several antenna types, array configurations and calibration techniques,
and pioneered analysis methods for the reconstruction of the most important
shower parameters, i.e., the arrival direction, the energy, and mass-dependent
observables such as the position of the shower maximum. In addition to a review
and update of previously published results, we also present new results based
on end-to-end simulations including all known instrumental properties. For
this, we applied the detector response to radio signals simulated with the
CoREAS extension of CORSIKA, and analyzed them in the same way as measured
data. Thus, we were able to study the detector performance more accurately than
before, including some previously inaccessible features such as the impact of
noise on the interferometric cross-correlation beam. These results led to
several improvements, which are documented in this paper and can provide useful
input for the design of future cosmic-ray experiments based on the digital
radio-detection technique.
