Construction of covering maps between orientable surfaces

Consider the following problem. Can a set of numbers be realized as boundar y covering indices of a covering map between surfaces? How to realize them? A set of equivalent criteria for this problem are found, which can be chec ked by a finite algorithm. Furthermore, the algorithm will also construct a solution if such one exists. As an application, a well-known group of nece ssary conditions are shown to be not sufficient in infinitely many cases, w hile in most cases, numbers satisfying them are realizable.