Enumeration and coding methods for a class of permutations and reversible logical gates

We introduce a great variety of coding methods for Boolean sparse invertible matrices and we use these methods to create a variety of bijections on the permutation group P(m) of the set {1,2,...,m}. Also, we propose methods for coding, enumerating and shuffling the set{0,...,2m?1}, i.e. the set of all m-bit binary arrays. Moreover we show that several well known reversible logic gates/circuits (on m-bit binary arrays) can be coded by sparse matrices.