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Hierarchical Locally Recoverable Codes on surfaces

Carolina Araujo, Luana Costa, Beth Malmskog, Jorge Mello, Eliza Menezes, Cecília Salgado, Lara Vicino

Abstract

We construct locally recoverable codes with hierarchy from surfaces in $\mathbb{A}^3$ admitting a fibration by curves of Artin-Schreier or Kummer type. We derive the parameters of our codes by leveraging the geometry and arithmetic of the fibration, which is obtained by projection onto one of the coordinates. As a byproduct, we obtain estimates for (and in one case an explicit count of) the number of rational points in certain families of surfaces.

Hierarchical Locally Recoverable Codes on surfaces

Abstract

We construct locally recoverable codes with hierarchy from surfaces in admitting a fibration by curves of Artin-Schreier or Kummer type. We derive the parameters of our codes by leveraging the geometry and arithmetic of the fibration, which is obtained by projection onto one of the coordinates. As a byproduct, we obtain estimates for (and in one case an explicit count of) the number of rational points in certain families of surfaces.
Paper Structure (15 sections, 15 theorems, 67 equations, 1 figure)

This paper contains 15 sections, 15 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of the article
  3. Notations and background
  4. Codes
  5. Algebraic geometric tools
  6. Artin-Schreier and Kummer Covers
  7. Codes on Artin-Schreier surfaces
  8. Code construction: a quasi-projective approach
  9. Dimension, locality and lower bounds for the minimum distance
  10. Examples of HLRCs from Artin-Schreier surfaces
  11. Codes on Kummer-type surfaces
  12. Code construction
  13. Dimension, locality and lower bounds for the minimum distance
  14. Examples of HLRCs from Kummer-type surfaces
  15. Observations, Generalizations, and Next Steps

Key Result

Theorem 2.11

If $C$ and $D$ are plane curves in $\mathbb{P}^2(K)$ given by homogeneous polynomials $g$ and $h$ without a common factor, then $i(C,D)=(\deg g)(\deg h).$

Figures (1)

  • Figure 1: The leftmost diagram summarizes the rational maps between the curves $\mathcal{H}_q$, $\overline{\mathcal{Z}}_\gamma$ and $\mathcal{Y}_\gamma$. The map $\pi_x$ is the projection onto the $x$-coordinate, the map $\varphi_\gamma$ is the triple cover defined above, and the horizontal dashed arrow represents the birational equivalence of $\overline{\mathcal{Z}}_\gamma$ and $\mathcal{Y}_\gamma$. The dotted arrows represent induced rational maps in the commutative diagram. The rightmost diagram is the "translation", in the function field setting, of the leftmost diagram.

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: PKLK12
  • Definition 2.5: sasidharan2015codes
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 49 more