For a given set \(S\subseteq \mathbb {Z}_m\) and \(\overline{n}\in \mathbb {Z}_m\), \(R_S(\overline{n})\) is defined as the number of solutions of the equation \(\overline{n}=\overline{s}+\overline{s'}\) with unordered pair \((\overline{s},\overline{s'})\in S^2\) and \(\overline{s}\ne \overline{s'}\). In this paper, we prove that if \(m=2^id,\,i\ge 1,\,2\not \mid d, d > 1\) then there exist two sets \(A,B\subseteq \mathbb {Z}_{m}\) with \(A\cup B=\mathbb {Z}_{m}\), \(A\cap B=\{\overline{r_1}<\overline{r_2}\}\), \(B\ne A+\overline{m/2}\) and \(R_{A}(\overline{n})=R_{B}(\overline{n})\) for all \(\overline{n}\in \mathbb {Z}_{m}\) if and only if there exists an odd integer \(u\ne d,~0<u<2d\) such that \(\overline{r_2}-\overline{r_1}=\overline{2^{i-1}u}\).

对于给定的集合\(S\subseteq \mathbb {Z}_m\)和\(\overline{n}\in \mathbb {Z}_m\)，\(R_S(\overline{n})\)被定义作为方程\(\overline{n}=\overline{s}+\overline{s'}\)与无序对\((\overline{s},\overline{s'})\ 的解数在 S^2\)和\(\overline{s}\ne \overline{s'}\)。在本文中，我们证明如果\(m=2^id,\,i\ge 1,\,2\not \mid d, d > 1\)那么存在两个集合\(A,B\subseteq \ mathbb {Z}_{m}\)与\(A\cup B=\mathbb {Z}_{m}\) , \(A\cap B=\{\overline{r_1}<\overline{r_2} \}\) , \(B\ne A+\overline{m/2}\)和\(R_{A}(\overline{n})=R_{B}(\overline{n})\)对于所有\(\overline{n}\in \mathbb {Z}_{m}\)如果并且仅当存在奇数整数\(u\ne d,~0<u<2d\)使得\(\overline{r_2}-\overline{r_1}=\overline{2^{i-1}u }\)。