Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hermitian-Lifted Codes

Hiram H. López, Beth Malmskog, Gretchen L Matthews, Fernando Piñero-González, Mary Wootters

TL;DR

The paper introduces Hermitian-Lifted Codes, a new class of curve-lifted locally recoverable codes defined via evaluation on the Hermitian curve $\mathcal{H}_q$ over $\mathbb{F}_{q^2}$. By restricting the allowed bivariate polynomials to those whose line-restrictions coincide with low-degree univariate polynomials on $\mathcal{H}_q$, the authors guarantee locality $r=q$ and availability $t=q^2-1$, with a rate that remains bounded away from zero as $q$ grows. The main result proves a positive lower bound on the rate, established through a detailed analysis of monomials that behave well under restriction modulo polynomials associated with lines on the curve; they prove a concrete bound of at least $0.007$ for $q=2^\ell$, $\ell\ge2$, and provide a counting argument for many good monomials. The paper further demonstrates the practical potential via explicit examples for $q=4,8,16,32$, where the Hermitian-Lifted Codes outperform the corresponding one-point Hermitian codes in rate, illustrating a new paradigm—curve-lifted codes—that could inspire broader constructions and improved parameters in the future.

Abstract

In this paper, we construct codes for local recovery of erasures with high availability and constant-bounded rate from the Hermitian curve. These new codes, called Hermitian-lifted codes, are evaluation codes with evaluation set being the set of $\mathbb{F}_{q^2}$-rational points on the affine curve. The novelty is in terms of the functions to be evaluated; they are a special set of monomials which restrict to low degree polynomials on lines intersected with the Hermitian curve. As a result, the positions corresponding to points on any line through a given point act as a recovery set for the position corresponding to that point.

Hermitian-Lifted Codes

TL;DR

The paper introduces Hermitian-Lifted Codes, a new class of curve-lifted locally recoverable codes defined via evaluation on the Hermitian curve over . By restricting the allowed bivariate polynomials to those whose line-restrictions coincide with low-degree univariate polynomials on , the authors guarantee locality and availability , with a rate that remains bounded away from zero as grows. The main result proves a positive lower bound on the rate, established through a detailed analysis of monomials that behave well under restriction modulo polynomials associated with lines on the curve; they prove a concrete bound of at least for , , and provide a counting argument for many good monomials. The paper further demonstrates the practical potential via explicit examples for , where the Hermitian-Lifted Codes outperform the corresponding one-point Hermitian codes in rate, illustrating a new paradigm—curve-lifted codes—that could inspire broader constructions and improved parameters in the future.

Abstract

In this paper, we construct codes for local recovery of erasures with high availability and constant-bounded rate from the Hermitian curve. These new codes, called Hermitian-lifted codes, are evaluation codes with evaluation set being the set of -rational points on the affine curve. The novelty is in terms of the functions to be evaluated; they are a special set of monomials which restrict to low degree polynomials on lines intersected with the Hermitian curve. As a result, the positions corresponding to points on any line through a given point act as a recovery set for the position corresponding to that point.

Paper Structure

This paper contains 15 sections, 8 theorems, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Hermitian Curve and Prior Code Constructions
  3. The Hermitian curve
  4. Evaluation codes and one-point codes on Hermitian curves
  5. Contrast of Hermitian-Lifted Codes with related literature
  6. Code Construction
  7. Main Theorem and Proof
  8. Behavior of monomials $t^k$ modulo $p_{\alpha, \beta}(t)$
  9. Bound on the rate of the code
  10. Examples
  11. The code $\mathcal{C}$ when $q=4$
  12. The code $\mathcal{C}$ when $q=8$
  13. The code $\mathcal{C}$ when $q=16$
  14. The code $\mathcal{C}$ when $q=32$
  15. Conclusion

Key Result

Theorem 4

Suppose that $q \geq 4$ is a power of $2$, and let $\mathcal{C}$ be as in Definition def:C. Then the rate of $\mathcal{C}$ is at least $0.007$.

Figures (4)

  • Figure 1: Exponent pairs $(a,b)$ with $x^ay^b\in \mathcal{C}$ for $q=4$ ($a$ is on horizontal axis).
  • Figure 2: Exponent pairs $(a,b)$ with $x^ay^b\in \mathcal{C}$ for $q=8$ ($a$ is on horizontal axis).
  • Figure 3: Exponent pairs $(a,b)$ with $x^ay^b\in \mathcal{C}$ for $q=16$ ($a$ is on horizontal axis).
  • Figure 4: Exponent pairs $(a,b)$ with $x^ay^b\in \mathcal{C}$ for $q=32$ ($a$ is on horizontal axis).

Theorems & Definitions (28)

  • Definition 1
  • proof
  • Definition 2
  • Definition 3
  • Definition 4: Hermitian-Lifted Codes
  • Remark 1: Horizontal lines
  • proof
  • Theorem 4
  • Proposition 5
  • proof
  • ...and 18 more