Multidimensional linear phase perfect reconstruction filter bank (MDLPPRFB) can be designed and implemented via lattice structure. The lattice structure for the MDLPPRFB with filter support N(MΞ) has been published by Muramatsu , where M is the decimation matrix, Ξ is a positive integer diagonal matrix, and N(N) denotes the set of integer vectors in the fundamental parallelepiped of the matrix N. Obviously, if Ξ is chosen to be other positive diagonal matrices instead of only positive integer ones, the corresponding lattice structure would provide more choices of filter banks, offering better trade-off between filter support and filter performance. We call such resulted filter bank as generalized-support MDLPPRFB (GSMDLPPRFB). The lattice structure for GSMDLPPRFB, however, cannot be designed by simply generalizing the process that Muramatsu employed. Furthermore, the related theories to assist the design also become different from those used by Muramatsu . Such issues will be addressed in this paper. To guide the design of GSMDLPPRFB, the necessary and sufficient conditions are established for a generalized-support multidimensional filter bank to be linear-phase. To determine the cases we can find a GSMDLPPRFB, the necessary conditions about the existence of it are proposed to be related with filter support and symmetry polarity (i.e., the number of symmetric filters ns and antisymmetric filters na). Based on a process (different from the one Muramatsu used) that combines several polyphase matrices to construct the starting block, one of the core building blocks of lattice structure, the lattice structure for GSMDLPPRFB is developed and shown to be minimal. Additionally, the result in this paper includes Muramatsu's as a special case.