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 $s \ge 4$.