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Dual of Algebraic Geometry codes from Hirzebruch surfaces

Alix Barraud

Abstract

In this paper, we give an explicit form for the dual of the algebraic geometry code $C_e(a,b)$ defined on an Hirzebruch surface $\mathcal{H}_e$ and parametrized by the divisor $aS_e + bF_e$, where $a,b\in\mathbb{N}$ and $S_e$ and $F_e$ generate the Picard group $\mathrm{Pic}( \mathcal{H}_e)$. Notably, we compute a lower bound for the minimum distance of $C_e(a,b)^\perp$. One of the main ingredient for our study is a new explicit form of the code $C_e(a,b)$ which we provide at the beginning of the paper. We also investigate some puncturing of $C_e(a,b)$, recovering other previously studied AG codes from toric surfaces. Finally, we provide a sufficient condition for orthogonal inclusions between the codes $C_e(a,b)$, and construct CSS quantum codes from them.

Dual of Algebraic Geometry codes from Hirzebruch surfaces

Abstract

In this paper, we give an explicit form for the dual of the algebraic geometry code defined on an Hirzebruch surface and parametrized by the divisor , where and and generate the Picard group . Notably, we compute a lower bound for the minimum distance of . One of the main ingredient for our study is a new explicit form of the code which we provide at the beginning of the paper. We also investigate some puncturing of , recovering other previously studied AG codes from toric surfaces. Finally, we provide a sufficient condition for orthogonal inclusions between the codes , and construct CSS quantum codes from them.

Paper Structure

This paper contains 30 sections, 33 theorems, 137 equations, 4 figures.

Table of Contents

  1. Basic properties of Hirzebruch surfaces
  2. Algebraic surfaces and their geometry
  3. Divisors on a surface
  4. Linear equivalence and Riemann--Roch space
  5. Intersection number
  6. Description of $\mathcal{H}_e$
  7. Riemann--Roch spaces of $\mathcal{H}_e$
  8. Algebraic Geometry codes on $\mathcal{H}_e$ and their dual distance
  9. Context
  10. Codes
  11. AG codes on a surface
  12. Evaluation morphism
  13. Parameters of $C_e(a,b)$
  14. Dual distance of $C_e(a,b)$
  15. Check-product of codes
  16. ...and 15 more sections

Key Result

Proposition 1.3

Let $a,b \in \mathbb{N}$. The Riemann--Roch space $L(aS_e+bF_e)$ is isomorphic to the $\mathop{\mathrm{\mathbb{F}}}\nolimits_q$--vector space denoted $L_* (aS_e + bF_e)$ generated by all monomials $M \in \mathop{\mathrm{\mathbb{F}}}\nolimits_q[X_1,X_2,T_1,T_2]$ of the form $M=X_1^{d_1}X_2^{d_2}T_1^{ The space $L_*(aS_e+bF_e)$ is explicitely given by

Figures (4)

  • Figure 1: The surfaces $\mathcal{H}_0$ and $\mathcal{H}_1$.
  • Figure 2: Construction of $\mathcal{H}_{e+1}$ from $\mathcal{H}_e$.
  • Figure 3: The fibres $F'_e$ and $F_e$, and the sections $S_e$ and $\sigma_e \simeq S_e + eF_e$.
  • Figure 4: Codeword associated to the monomial $M=T_1^{c_1}T_2^{c_2}X_1^{d_1}X_2^{d_2}$.

Theorems & Definitions (78)

  • Remark 1.1
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • proof
  • Remark 2.1
  • Definition 2.4: The code $C_e(a,b)$
  • Definition 2.6: Reed--Solomon codes
  • Proposition 2.7: Stic09
  • Proposition 2.8
  • ...and 68 more