Codes from infinitely near points

Abstract

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 $\mathbf{F}_q$. An infinitely near point of order $\mu$ is a point $P$ on a variety $X^\prime$ obtained by $\mu$ 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 $\mu = 0$). Given a set $S$ of infinitely near points $\left\{P_1, \ldots , P_n \right\}$, we associate to $f$ its set of values $(f(P_1), \ldots, f(P_n))$ in $\mathbf{F}^n_q$. 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 $V \to \mathbf{F}^n_q$, 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 $\mathcal{S}$. 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.