## Asian Journal of Mathematics

- Asian J. Math.
- Volume 16, Number 1 (2012), 89-102.

### Codes from infinitely near points

Bruce M. Bennett, Hing Sun Luk, and Stephen S.-T. Yau

#### 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.

#### Article information

**Source**

Asian J. Math., Volume 16, Number 1 (2012), 89-102.

**Dates**

First available in Project Euclid: 13 March 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ajm/1331663452

**Mathematical Reviews number (MathSciNet)**

MR2904913

**Zentralblatt MATH identifier**

1288.94111

**Subjects**

Primary: 94B27: Geometric methods (including applications of algebraic geometry) [See also 11T71, 14G50]

**Keywords**

Algebraic-geometry code blowing up infinitely near points

#### Citation

Bennett, Bruce M.; Luk, Hing Sun; Yau, Stephen S.-T. Codes from infinitely near points. Asian J. Math. 16 (2012), no. 1, 89--102. https://projecteuclid.org/euclid.ajm/1331663452