Let be an integer such that and set with cardinality. The goal of this paper is to present a new way to encoding all subset of the set by sub-exceeding function on. After giving some proprieties of the set of sub-exceeding functions on we present a particular subset which we denote by such that his cardinality is. Since this subset has the same number of elements as the set of parts of denoted (), so we can construct a bijective map denoted from the set to (). As results, given an arbitrary

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... et of , we can encode it by using a sub-exceeding function in. Reciprocally, if we have a sub-exceeding function in , we can decode it and find the subset of that it represents. Moreover, we can also define in the equivalent of the usual laws and which operate in ().