math.CO — Combinatorics

Complete planar arcs over nonsquare binary fields

Contributed by Jose Voloch

Let $m'_2(q)$ be the largest size of a complete arc in $\operatorname{PG}(2,q)$ other than a hyperoval. We prove $m'_2(q) \le q-\floor{q^{2/3}-q^{1/3}}, q=2^h\ge512,\ h\text{ odd}$. The geometric statement underlying the proof is a degree obstruction for arbitrary geometrically irreducible Frobenius nonclassical plane curves; an origin from an arc is unnecessary. The argument uses the curve and a Frobenius twist of its dual, exact weighted point counts, and a local conductor inequality that allows arbitrary singularities. We give a sharper bound obtained by optimizing over powers of two, determine its leading constant according to $h\bmod3$, and prove further refinements using a characteristic-two normal form. Exact finite computations give, for example, $m'_2(128)\le105$, $m'_2(512)\le452$, and $m'_2(2048)\le1875$.