Lineaarsetel

Lineaarsetel is a concept in finite geometry that describes certain structured subsets of projective spaces, typically denoted as linear sets in PG(n-1, q^t). The term is used in contexts where the classical idea of a subspace is generalized by incorporating a field extension. In the standard construction, one fixes a finite field F_q and its extension F_{q^t}, together with a positive integer n. Consider the projective space PG(n-1, q^t). A lineaarsetel L_U is obtained from an F_q-subspace U of F_{q^t}^n by taking as points the projective images of all nonzero vectors in U. The rank of the linear set is k = dim_{F_q}(U); equivalently, L_U is said to have rank k.

Construction and basic properties. If U is an F_q-subspace of F_{q^t}^n with dimension k, then L_U consists of the points of PG(n-1, q^t) that can be represented by nonzero vectors of U. The set L_U is invariant under the natural action of the multiplicative group of F_q and is therefore considered F_q-rational. A standard bound for the size of a lineaarsetel is |L_U| ≤ (q^k − 1)/(q − 1), with equality in certain extremal cases. The rank k and the ambient dimension n place structural constraints on L_U, and different choices of U yield a variety of linear sets with distinct geometric and combinatorial properties.

Special cases and terminology. If k = n, the linear set often corresponds to a subgeometry PG(n-1, q) embedded in PG(n-1, q^t). A notable class is that of scattered linear sets, where each point of L_U has minimal weight, a condition with relevance to blocking sets and spreads in finite geometry. Linear sets also appear in coding theory and network coding through their connections to projective systems and rank-metric codes.

Applications and related notions. Lineaarsetel provide a framework for constructing and analyzing families of subsets in projective spaces with controlled intersection patterns. They are related to field reduction, clubs, and various families of linear blocking sets, and they inform methods in finite geometry and related coding-theory constructions.