On translation hyperovals in semifield planes

The article delves into the intricate world of translation hyperovals in semifield planes, focusing on their existence and classification. It begins by introducing hyperovals as extremal combinatorial objects in projective planes and their significance in Desarguesian and non-Desarguesian planes. The main objective is to disprove Cherowitzo's conjecture, which posits that translation hyperovals exist in all translation planes of even order. The authors achieve this by demonstrating that the twisted field plane of order 64, a semifield plane, contains no translation hyperovals. This discovery has broader implications for the existence of scattered subspaces and the covering radius of rank-metric codes. The article also explores the connections between hyperovals, scattered subspaces, and MRD codes, highlighting the challenges and open questions in this complex field.