Codes from infinitely near points
Bruce M. Bennett, Hing Sun Luk, Stephen S.‐T. Yau · Asian Journal of Mathematics · 2012
We introduce a new class of nonlinear algebraic-geometry codes based on evaluation of functions on infinitely near points.Let X be an algebraic variety over the finite field F q .An infinitely near point of order µ is a point P on a variety X ′ obtained by µ iterated blowing-ups starting from X.Given such a point P and a function f on X, we give a definition of f (P ) which is nonlinear in f (unless µ = 0).Given a set S of infinitely near points {P 1 , • • • , Pn}, we associate to f its set of values (f (P 1 ), • • • , f (Pn)) in F n q .Let V be a k dimensional vector space of functions on X. Evaluation of functions in V at the n points of S gives a map V → F n q , which we view as an (n, q k , d) code when the map is injective.Here d is the largest integer such that a function in V is uniquely determined by its values on any n -d + 1 points of S.These codes generalize the Reed-Solomon codes, but unlike the R-S codes they can be constructed to have arbitrarily large code length n.The first nontrivial case is where X = A 2 F q , affine 2-space, and we study this case in detail.