Burniat-type surfaces and a new family of surfaces with p_g = 0, K^2 = 3

In this paper, one of a series devoted to the classification, the moduli spaces and the discovery of new surfaces of general type with geometric genus p_g= 0, we generalize a classical construction method due to Burniat (and revisited by Inoue), constructing surfaces with K^2 = 3 and a with a new fundamental group of order 16.