Linearsete

Linearsete, in English often called linear sets, are objects in finite geometry arising from subspaces defined over a subfield of a finite extension field. They provide a way to describe families of points in a projective space by lifting a smaller field structure into a larger one. The term is commonly used in contexts involving projective spaces over extension fields and subspaces over the base field.

Formally, let q be a prime power and F_q its finite field, with F_{q^n} an extension field. Consider the projective space PG(r−1, q^n). Take an F_q-subspace U of the vector space F_{q^n}^r with dimension k (the rank of the linearset). The linearset L(U) is the set of all points ⟨u⟩ in PG(r−1, q^n) with u ≠ 0 and u ∈ U. Thus L(U) = {⟨u⟩_{q^n} : u ∈ U \ {0}}. The rank k determines the size and combinatorial properties of the linearset, though the exact number of points can vary with U.

Key properties include invariance under certain linear transformations and the existence of special families, such as scattered linear sets, where L(U) attains large size relative to k. Linearsets also relate to constructions of spreads, subgeometries, and rank-metric codes, making them relevant in coding theory and finite geometry.

Historically, linearsets have been studied by researchers in finite geometry, with developments linking them to applications in network coding and geometric constructions. They are a standard topic in the study of how subfield structures interact with projective spaces.