Captain reported the inability to ascertain the status of taxiway closures at TSP airport caused concern regarding the suitability of taxiways while operating at the airport.

Narrative

When planning the flight into TSP; the NOTAM revealed several taxiways that were closed. The day of the flight; we attempted to contact the airport authority. It was Day 0 and the airport authority did not answer the phone; however; the Tehachapi police department answered the call. They had no knowledge of the airfield situation.The notes revealed several taxiways that were closed; and upon reviewing the Tehachapi plates; there was only one plate for the airfield diagram; as there are no approaches to the runway. The airfield diagram plate only showed the runway and taxiways; but no taxiway designators. That is why we wanted to contact the airfield authority to determine the suitability of the field.We launched for Tehachapi and overflew the airport and viewed the construction and determined it was safe to land.Upon clearing the runway; we noticed that down the parallel taxiway they had orange cones on the right side and on the left side; they had new taxiway designators that were fixed as it all airports.We slowly taxi down the parallel taxiway; and the cones were extremely close to the wing however; they were lower than the wing itself as were the taxiway designators taxi into parking and when we left; we chose a different route to get to the runway for departure.In light of the inability to talk to airfield personnel; I would highly recommend we do not use Tehachapi airport until all of the construction is complete. There are just too many hazardous areas to navigate around; and without adequate reference to taxiways on the only airfield diagram that does not have taxiway designation; I recommend we suspend operations into TSP until construction is complete.Cause: Lack of ability to talk to airport personnel and an inadequate airfield diagram that identifies taxiways.Suggestions: Better communications with airport personnel; especially during construction operations.