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Perfect weighted coverings of radius one have been studied in the Hamming metric and in the Lee metric. For practical reasons, we present them in a slightly different way, yet equivalent. Given an integer k, the k-neighborhood of an element is the set of elements at distance at most k. Let a and b be two integers. An (a,b)-code is a set of elements such that elements in the code have a+1 elements belonging to the code in their k-neighborhood and other elements have b elements belonging to the code in their k-neighborhood. In this paper, we study the (a,b)-codes in Z/nZ, where the distance between x and y is |x-y|mod[n] and we characterize the existence of a non trivial (a,b)-code in Z/nZ. |
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