On the classification of product-quotient surfaces with $q=0$, $p_{g}=3$ and their canonical map

In this work, we present new results to produce an algorithm that returns, for any fixed pair of positive integers K^{2} and \chi , all regular surfaces S of general type with self-intersection of the canonical class K_{S}^{2}=K^{2} and Euler characteristic \chi(\mathcal{O}_{S})=\chi , which are product-quotient surfaces. The key result we obtain is an algebraic characterization of all families of regular product-quotients surfaces, up to isomorphism, arising from a pair of G -coverings of \mathbb{P}^{1} . As a consequence of our work, we provide a classification of all regular product-quotient surfaces S of general type with 23\leq K_{S}^{2}\leq 32 and \chi(\mathcal{O}_{S})=4 . Furthermore, we study their canonical map and present several new examples of surfaces of general type with a high degree of the canonical map.