A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with \(n=2^s\) sides are unknown when \(s \ge 4\). In this paper, we construct a family of convex small \(n\)-gons, \(n=2^s\) with \(s\ge 4\), and show that their perimeters and their widths are within \(O(1/n^8)\) and \(O(1/n^5)\) of the maximal perimeter and the maximal width, respectively. From this result, it follows that Mossinghoff's conjecture on the diameter graph of a convex small \(2^s\)-gon with maximal perimeter is not true when \(s \ge 4\).