On circulant thin Lehman matrices

In 1998, C. Lütolf and F. Margot [C. Lütolf, F. Margot, A Catalog of minimally nonideal matrices, Math. Methods Oper. Res. 47 (1998) 221–241] defined the notion of Lehman matrices in order to classify minimally non-ideal clutters. In the paper [H. Shinohara, Thin Lehman Matrices Arising from Finite Groups, Linear Algebra and Appl. to appear], one of the authors defined the notion of 1-overlapped factorizations of finite group which produces an infinite family of Lehman matrices and proved many properties of 1-overlapped factors. In this paper, we characterize circulant thin Lehman matrices with small constant line sum and provide many useful properties of 1-overlapped factors of cyclic groups.