Existence of Ideal Matrices

An \(m \times n\) ideal matrix is a \(3\)-periodic \(m \times n\) binary matrix which satisfies the following two conditions: (1) each column of this matrix contains precisely one \(1\) and (2) if it is visualized as a dot pattern (with each dot representing a \(1\)), then the number of overlapping dots at all actual shifts are \(1\) or \(0\). Let \(s(n)\) denote the smallest integer \(m\) such that an \(m \times n\) ideal matrix exists. In this paper, we reduce the upper bound of \(s(n)\) which was found by Fung, Siu and Ma. Also, we list an upper bound of \(s(n)\) for \(14 \leq n \leq 100\).