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On QC and GQC algebraic geometry codes

Matteo Bonini, Arianna Dionigi, Francesco Ghiandoni

TL;DR

This work generalizes QC and GQC algebraic-geometry code constructions beyond elliptic curves by leveraging Kummer-type and other high-order automorphisms on hyperelliptic, norm-trace, and Hermitian curves. It provides a unified AG framework that uses automorphism orbits on rational points to build codes $\mathcal C(D_1,D_2)$ with co-index equal to the order of the acting automorphism, together with explicit divisors and parameter formulas. The paper analyzes several curve families, including maximal curves over $\bF_{q^2}$, to obtain asymptotically good parameters and flexible co-index choices; it also gives concrete examples and orbit structures to realize QC/GQC codes with diverse lengths and distances. Overall, the approach broadens the design space for QC/GQC codes via algebraic-geometry methods, offering new codes with practical encoding/decoding properties and potential implications for post-quantum cryptography.

Abstract

We present new constructions of quasi-cyclic (QC) and generalized quasi-cyclic (GQC) codes from algebraic curves. Unlike previous approaches based on elliptic curves, our method applies to curves that are Kummer extensions of the rational function field, including hyperelliptic, norm-trace, and Hermitian curves. This allows QC codes with flexible co-index. Explicit parameter formulas are derived using known automorphism-group classifications.

On QC and GQC algebraic geometry codes

TL;DR

This work generalizes QC and GQC algebraic-geometry code constructions beyond elliptic curves by leveraging Kummer-type and other high-order automorphisms on hyperelliptic, norm-trace, and Hermitian curves. It provides a unified AG framework that uses automorphism orbits on rational points to build codes with co-index equal to the order of the acting automorphism, together with explicit divisors and parameter formulas. The paper analyzes several curve families, including maximal curves over , to obtain asymptotically good parameters and flexible co-index choices; it also gives concrete examples and orbit structures to realize QC/GQC codes with diverse lengths and distances. Overall, the approach broadens the design space for QC/GQC codes via algebraic-geometry methods, offering new codes with practical encoding/decoding properties and potential implications for post-quantum cryptography.

Abstract

We present new constructions of quasi-cyclic (QC) and generalized quasi-cyclic (GQC) codes from algebraic curves. Unlike previous approaches based on elliptic curves, our method applies to curves that are Kummer extensions of the rational function field, including hyperelliptic, norm-trace, and Hermitian curves. This allows QC codes with flexible co-index. Explicit parameter formulas are derived using known automorphism-group classifications.
Paper Structure (12 sections, 13 theorems, 73 equations)

This paper contains 12 sections, 13 theorems, 73 equations.

Table of Contents

  1. Preliminaries
  2. Algebraic curves
  3. Linear Codes
  4. Algebraic Geometry Codes
  5. Construction of (generalized) quasi-cyclic AG codes
  6. Curves defined by separated polynomials
  7. Hyperelliptic curves
  8. Genus 2
  9. The $\mathbb{F}_{q^2}$-maximal curve $x^2=y^{2g+1}+1$
  10. The norm-trace curve
  11. The Hermitian curve
  12. A quotient of the Hermitian curve with a large cyclic automorphism group

Key Result

Theorem 1.1

Let $H$ be an abelian subgroup of $\mathrm{Aut}_\mathbb{K}(\mathcal{X})$. Then

Theorems & Definitions (30)

  • Theorem 1.1: HKT
  • Proposition 1.2: Singleton bound
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 3.1
  • ...and 20 more