Some recent results in finite geometry and coding theory arising from the Gale transform

The Gale transform is an involution on sets of points in a projective space. It plays a crucial role in several different subjects, such as algebraic geometry, optimization, coding theory, and so on. We give a brief survey—from a finite geometry point of view—on the algebraic and geometrical implications of the Gale transform with emphasis on its applications to coding theory, and describe some recent results. 1 – Introduction The Gale transform of a set T consisting of γ labelled points of a projective space PG(r, q) is an involution which maps T into a set T ′ consisting of γ labelled points of PG(s, q), defined up to automprphisms of PG(s, q), with γ = r+ s+2. The simplest way to define the Gale transform of a set of points is in terms of projective coordinates. Choose homogeneous coordinates in such a way that the coordinates of the points of T are the rows of the matrix ( Ir+1 A ) , where In denotes the n× n identity matrix and A is an (s+ 1)× (r + 1) matrix. Then, the Gale transform of T is the set T ′ consisting of the points of PG(s, q)