We solve the smallest four open cases of weighing matrices, for , which completes the existence question for weight 16. In addition we solve the open two-core matrix . There is a common theme for the construction of all such matrices, which is called here . 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.

中文翻译：

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通过分区组动作创建新的权重矩阵

我们解决了权重矩阵 的最小四个开放情况，这完成了权重 16 的存在性问题。此外，我们解决了开放的两核矩阵 。所有此类矩阵的构造都有一个共同的主题，此处称为 。划分群矩阵的研究概括了一些众所周知的结构，即有边界或无边界的一核和二核循环结构、分块循环矩阵、勒让德对等等。这种构造可推广到任意群并带有我们在某些情况下分析的代数结构。我们这里的方法可以适用于更大的称重矩阵。我们还添加了对同权重 5 的双芯称重矩阵中零点（晶体组）可能位置的完整分析。