Asian Journal of Mathematics

Codes from infinitely near points

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

Full-text: Open access

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


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