During an after landing taxi on a closed and unbarricaded taxiway, the pilot noticed a flickering landing gear indicator light. When the pilot diverted his attention to the indicator light, the airplane veered off the closed taxiway and into a drainage ditch that was under construction adjacent to the taxiway. The airplane sustained substantial damage, which included damage to the firewall, the right rear landing gear spar, and buckling of the right wing. The pilot was uninjured. A notice to airman (NOTAM) for the taxiway closure was issued and was also known by the pilot. The airport operations manager initially told Federal Aviation Administration (FAA) inspectors that there were no NOTAMs issued pertaining to the taxiway closure. Upon completion of the FAA's on-scene documentation, the airport manager was asked whether there was a NOTAM issued to which the airport operations manager stated that there was a NOTAM for general construction.