A Hadamard matrix is balanced splittable if some subset of its rows has the property that the dot product of every two distinct columns takes at most two values. This definition was introduced by Kharaghani and Suda in 2019, although equivalent formulations have been previously studied using different terminology. We collate previous results phrased in terms of balanced splittable Hadamard matrices, real flat equiangular tight frames, spherical two-distance sets, and two-distance tight frames. We use combinatorial analysis to restrict the parameters of a balanced splittable Hadamard matrix to lie in one of several classes, and obtain strong new constraints on their mutual relationships. An important consideration in determining these classes is whether the strongly regular graph associated with the balanced splittable Hadamard matrix is primitive or imprimitive. We construct new infinite families of balanced splittable Hadamard matrices in both the primitive and imprimitive cases. A rich source of examples is provided by packings of partial difference sets in elementary abelian 2-groups, from which we construct Hadamard matrices admitting a row decomposition so that the balanced splittable property holds simultaneously with respect to every union of the submatrices of the decomposition.

翻译：若Hadamard矩阵的某行子集满足任意两列的内积至多取两个值，则称该矩阵为平衡可分裂的。该定义由Kharaghani与Suda于2019年提出，尽管此前已有研究者使用不同术语研究了等价形式。我们梳理了以往用平衡可分裂Hadamard矩阵、实平坦等角紧框架、球面两点距离集及两点距离紧框架等术语表述的结果。通过组合分析，我们将平衡可分裂Hadamard矩阵的参数限制在若干类中，并对其相互关系建立了强有力的新约束。确定这些类别的重要考量因素在于：与平衡可分裂Hadamard矩阵关联的强正则图是原图还是非原图。我们在原图与非原图情形下均构造了无限新族平衡可分裂Hadamard矩阵。初等阿贝尔2-群中部分差集的堆积提供了丰富的例子来源，由此我们构造的Hadamard矩阵具有行分解结构，使得平衡可分裂性质同时适用于该分解的任意子矩阵并集。