The dual minimum distance of arbitrary-dimensional algebraic–geometric codes

In this article, the minimum distance of the dual C⊥ of a functional code C on an arbitrary-dimensional variety X over a finite field Fq is studied. The approach is based on problems à la Cayley–Bacharach and consists in describing the minimal configurations of points on X which fail to impose independent conditions on forms of some degree m. If X is a curve, the result improves in some situations the well-known Goppa designed distance.