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Parameterized affine codes

Hiram H. Lopez, Eliseo Sarmiento, Maria Vaz Pinto, Rafael H. Villarreal

Abstract

Let K be a finite field and let X* be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Groebner bases, to compute the length and the dimension of C_X*(d), the parameterized affine code of degree d on the set X*. If Y is the projective closure of X*, it is shown that C_X^*(d) has the same basic parameters that C_Y(d), the parameterized projective code on the set Y. If X* is an affine torus, we compute the basic parameters of C_X*(d). We show how to compute the vanishing ideals of X* and Y.

Parameterized affine codes

Abstract

Let K be a finite field and let X* be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Groebner bases, to compute the length and the dimension of C_X*(d), the parameterized affine code of degree d on the set X*. If Y is the projective closure of X*, it is shown that C_X^*(d) has the same basic parameters that C_Y(d), the parameterized projective code on the set Y. If X* is an affine torus, we compute the basic parameters of C_X*(d). We show how to compute the vanishing ideals of X* and Y.

Paper Structure

This paper contains 3 sections, 13 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Computing the length and dimension of an affine parameterized code
  3. Computing the dimension and length of $C_{X^*}(d)$

Key Result

Proposition 2.3

(harris, geramita-cayley-bacharach) $h_Y(d)=|Y|$ for $d\geq |Y|-1$.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6
  • proof
  • Corollary 2.7
  • proof
  • ...and 19 more