New weighing matrices via partitioned group actions

We solve the smallest four open cases of weighing matrices, W(n,16) for n=23,25,27,29, which completes the existence question for weight 16. In addition we solve the open two-core matrix W(102,97). There is a common theme for the construction of all such matrices, which is called here partitioned group matrices. The study of partitioned group matrices generalizes some well known constructions, namely the one-core and two-core circulant constructions, with or without borders, block circulant matrices, Legendre pairs and many more. Such constructions generalize to arbitrary groups and carry an algebraic structure which we analyze in some cases. Our methods here can be made practical for larger weighing matrices. We also add a complete analysis of the possible location of the zeros (crystal sets) in two-core weighing matrices of co-weight 5.