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Configurations of subspaces like equichordal and equiisoclinic tight fusionframes, which are in some sense optimally spread apart and which also havereconstruction properties emulating those of orthonormal bases, are useful invarious applications, such as wireless communications and quantum informationtheory. In this paper, a new construction of infinite classes of equichordaltight fusion frames built on semiregular divisible difference sets ispresented. Sometimes this construction yields an equiisoclinic packing. Each ofthe constructed fusion frames is shown to have both a flat representation and asparse representation. Furthermore, integrality conditions which characterizewhen equichordal and equiisoclinic fusion frames can have orthonormal baseswith entries in a subring of the algebraic integers are proven. Keywords:fusion frame, Grassmannian packing, difference sets, simplex bound,equichordal, equiisoclinic

New Constructions and Characterizations of Flat and Almost Flat Grassmannian Fusion Frames