The number of rational points on hyperelliptic supersingular curves of genus 4 in characteristic 2

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 hyperelliptic supersingular curves of genus 4 over finite fields of order 2n, n odd.