Hyperelliptic curves in characteristic 2

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In this paper we prove that there are no hyperelliptic supersingular curves over F₂bar of genus 2ⁿ-1 for any integer n>1. Let g be a natural number, and h=floor(log₂(g+1)+1). Let X be a hyperelliptic curve over F₂bar of genus g>2 and 2-rank zero, given by an affine equation y²-y=c₂g₊₁ x²ᵍ⁺¹ +...+ c₁ x. We prove that the first slope of the Newton polygon of X is bigger than or equal to 1/h. We also prove that the equality holds if (I) g<2ʰ-2, c₂ₕ₋₁ is nonzero; or (II) g=2ʰ-2, c₂ₕ₋₁ or c₃₍₂ʰ⁻¹₎₋₁ is nonzero. We prove that genus-4 hyperelliptic curve over F₂bar are precisely those with equations y² - y = x⁹ + a x⁵ + b x³.