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New classes of linearly independent (LI) transforms that possess fast forward and inverse butterfly diagrams and their corresponding polynomial expansions over Galois Field (2) [GF(2)] are introduced in this paper. The transforms have the smallest computational complexity among all known LI transforms and therefore can be calculated in shorter time when the computation is done by software. Alternatively, the transforms can also be calculated easily and efficiently using hardware. Here, the recursive definitions and fast transform calculations of four basic fastest LI transforms are first given. The definitions are then extended to generate a larger family of LI transforms with the same computational cost through reordering and permutation. Various properties of the transforms and relations between them are presented followed by their hardware implementations as well as experimental results for some binary benchmark functions.