  One of the big questions in the area of curves over finite fields concerns
the distribution of the numbers of points: Which numbers occur as the number of
points on a curve of genus $g$? The same question can be asked of various
subclasses of curves. In this article we classify the possibilities for the
number of points on genus 4 hyperelliptic supersingular curves over finite
fields of order $2^n$, $n$ odd.
