On translation hyperovals in semifield planes
Copyright policy )On translation hyperovals in semifield planes
In this paper we demonstrate the first example of a finite translation plane which does not contain a translation hyperoval, disproving a conjecture of Cherowitzo. The counterexample is a semifield plane, specifically a Generalised Twisted Field plane, of order $64$. We also relate this non-existence to the covering radius of two associated rank-metric codes, and the non-existence of scattered subspaces of maximum dimension with respect to the associated spread.
- University College Dublin Ireland
translation plane, Blocking sets, ovals, \(k\)-arcs, hyperoval, semifield, generalized twisted field, Semifields, Polynomials over finite fields, Nonassociative division algebras, Linear codes and caps in Galois spaces, MRD code, 12K10, 51E21, 51A40, Translation planes and spreads in linear incidence geometry, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), scattered subspace
translation plane, Blocking sets, ovals, \(k\)-arcs, hyperoval, semifield, generalized twisted field, Semifields, Polynomials over finite fields, Nonassociative division algebras, Linear codes and caps in Galois spaces, MRD code, 12K10, 51E21, 51A40, Translation planes and spreads in linear incidence geometry, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), scattered subspace
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