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Quantum codes from a new construction of self-orthogonal algebraic geometry codes

Fernando Hernando, Gary McGuire, Francisco Monserrat, Julio José Moyano-Fernández

TL;DR

This work develops a broad, unified method to construct quantum stabilizer codes from self-orthogonal algebraic geometry codes by exploiting a general divisor-theoretic framework. A key result (Theorem 2) expresses the dual of a one-point AG code in terms of a shifted divisor that involves a nonzero divisor $M$ and a derivative-based divisor $f_A$, enabling self-orthogonality for many curve families beyond Castle curves. The paper introduces explicit families of curves, including $A_{n,q,\\ell}$, $B_{q,G}$, and $C_{q,\\ell}$ (and their variants), derives tractable bounds for self-orthogonality under Euclidean and Hermitian inner products, and provides concrete quantum-code parameters that often violate the Gilbert–Varshamov bound. By demonstrating GV-exceeding quantum codes from a wide set of curves (with nonzero $M$ in many cases), the work significantly broadens the pool of usable AG codes for quantum error correction and highlights new pathways for code optimization in quantum information processing.

Abstract

We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes, which greatly extends the class of a previous paper due to Munuera, Tenório and Torres. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known.

Quantum codes from a new construction of self-orthogonal algebraic geometry codes

TL;DR

This work develops a broad, unified method to construct quantum stabilizer codes from self-orthogonal algebraic geometry codes by exploiting a general divisor-theoretic framework. A key result (Theorem 2) expresses the dual of a one-point AG code in terms of a shifted divisor that involves a nonzero divisor and a derivative-based divisor , enabling self-orthogonality for many curve families beyond Castle curves. The paper introduces explicit families of curves, including , , and (and their variants), derives tractable bounds for self-orthogonality under Euclidean and Hermitian inner products, and provides concrete quantum-code parameters that often violate the Gilbert–Varshamov bound. By demonstrating GV-exceeding quantum codes from a wide set of curves (with nonzero in many cases), the work significantly broadens the pool of usable AG codes for quantum error correction and highlights new pathways for code optimization in quantum information processing.

Abstract

We present new quantum codes with good parameters which are constructed from self-orthogonal algebraic geometry codes. Our method permits a wide class of curves to be used in the formation of these codes, which greatly extends the class of a previous paper due to Munuera, Tenório and Torres. These results demonstrate that there is a lot more scope for constructing self-orthogonal AG codes than was previously known.

Paper Structure

This paper contains 15 sections, 22 theorems, 61 equations, 1 table.

Table of Contents

  1. Introduction
  2. AG codes and their duals
  3. Main results
  4. Curves $A_{n,q,\ell}$
  5. Curves $B_{q,G}$
  6. Curves $C_{q,\ell}$
  7. Curves $C_{q, \ell}$
  8. Curves $C_{q^s,\ell}$
  9. Application to quantum codes
  10. Euclidean Inner Product
  11. Curves $A_{n,q,\ell}$
  12. Curves $B_{q,x^{q^k+1}+x}$
  13. Curves $C_{q,\ell}$
  14. Hermitian Inner Product
  15. Comparison with other papers

Key Result

Theorem 3.1

Let $C$ be a smooth affine plane curve over ${\bf F}$ with only one place at infinity. Let $g$ be the genus of $\tilde{\chi}_C$ and let Let $f_{\mathcal{A}}(z):=\prod_{a\in {\mathcal{A}}} (z-a)\in {\bf F}[z]$. Let $M$ be the divisor of affine zeroes of the rational function of $\tilde{\chi}_C$ defined by the derivative $f'_{\mathcal{A}}(z)$, as defined above. Then the following hold:

Theorems & Definitions (56)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • proof
  • Remark 3.4
  • ...and 46 more