Let \(G\) be a finite abelian group and \(S\) a sequence with elements of \(G\) . Let \(|S|\) denote the length of \(S\) and \(\mathrm{supp}(S)\) the set of all the distinct terms in \(S\) . Let \(\Sigma(S)\) denote the set of group elements which can be expressed as a sum of a nonempty subsequence of \(S\) . It is known that if \({0\notin\Sigma(S)}\) , then \(|\Sigma(S)| \geq |S|+|\mathrm{supp}(S)|-1\) . The sequence \(S\) satisfying \(0\not\in\Sigma(S)\) and \(|\Sigma(S)| = |S| + |\mathrm{supp}(S)| - 1\) has been described. In this paper, we determine the sequence \(S\) with \(0\not\in\Sigma(S)\) such that \(|\Sigma(S)|=|S|+|\mathrm{supp}(S)|\) . As a consequence, we obtain some new results on the sequence \(S\) with \(0\not\in\Sigma_{|G|}(S)\) and \(|\Sigma_{|G|}(S)|=|S|-|G|+|\mathrm{supp}(S)|-1\) , where \(\Sigma_{|G|}(S)\) denotes the set of group elements which can be expressed as a sum of a subsequence of \(S\) of length \(|G|\) .
