(a,b)(a,b)-codes in Z/nZZ/nZ

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 kk, the kk-neighborhood of an element is the set of elements at distance at most kk. Let aa and bb be two integers. An (a,b)(a,b)-code is a set of elements such that elements in the code have a+1a+1 elements belonging to the code in their kk-neighborhood and other elements have bb elements belonging to the code in their kk-neighborhood. In this paper, we study the (a,b)(a,b)-codes in Z/nZZ/nZ, where the distance between xx and yy is |x−y|mod[n]|x−y|mod[n] and we characterize the existence of a non trivial (a,b)(a,b)-code in Z/nZZ/nZ.