On generalized \(k\)-arcs in \( \mathbb{PG} (2n,q)\)
On generalized \(k\)-arcs in \( \mathbb{PG} (2n,q)\)
The notion of a generalized \(k\)-arc in \(\mathbb{PG}(2n,q)\) is introduced. When \(k = \frac{q^{n+1}-1}{q-1} +1\) it is demonstrated that the existence of a generalized \(k\)-arc in \( \mathbb{PG}(2n,q)\) leads to a construction of a partial geometry, a strongly regular graph and a two-weight code. Such \(k\)-arcs are called generalized hyperovals. It is proved that no such generalized hyperovals exist when \(q\) is odd. For each \(n\geq 2\) and \(q=2\) it is shown that each generalized hyperoval of \(\mathbb{PG}(2n,q)\) is a partition of \(\mathbb{PG}(2n,2) \setminus \mathbb{PG}(n,2)\). Related structures are also discussed.
\(k\)-arc, two-weight code, strongly regular graph, Combinatorial structures in finite projective spaces, Combinatorial aspects of finite geometries, hyperoval, partial geometry, Finite partial geometries (general), nets, partial spreads, Geometric methods (including applications of algebraic geometry) applied to coding theory
\(k\)-arc, two-weight code, strongly regular graph, Combinatorial structures in finite projective spaces, Combinatorial aspects of finite geometries, hyperoval, partial geometry, Finite partial geometries (general), nets, partial spreads, Geometric methods (including applications of algebraic geometry) applied to coding theory