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New Quantum Codes from Evaluation and Matrix-Product Codes

Carlos Galindo, Fernando Hernando, Diego Ruano

TL;DR

This work develops a versatile framework for constructing quantum stabilizer codes from algebraic codes, combining CSS, matrix-product codes, and Steane's enlargement. By leveraging three code families—Reed-Muller, hyperbolic, and affine variety codes—and their subfield-subcodes, the authors derive extensive families of stabilizer codes over various finite fields, with explicit parameters $[[n,k,d]]_q$. They demonstrate numerous improvements over existing records, including binary codes of length $127$ and $128$, and non-binary codes that meet or exceed BCH-based constructions and Gilbert-Varshamov bounds in several cases. The results provide a broad and practical toolkit for quantum error correction, with both theoretical parameters and concrete code tables applicable to quantum communication and computation.

Abstract

Stabilizer codes obtained via CSS code construction and Steane's enlargement of subfield-subcodes and matrix-product codes coming from generalized Reed-Muller, hyperbolic and affine variety codes are studied. Stabilizer codes with good quantum parameters are supplied, in particular, some binary codes of lengths 127 and 128 improve the parameters of the codes in http://www.codetables.de. Moreover, non-binary codes are presented either with parameters better than or equal to the quantum codes obtained from BCH codes by La Guardia or with lengths that can not be reached by them.

New Quantum Codes from Evaluation and Matrix-Product Codes

TL;DR

This work develops a versatile framework for constructing quantum stabilizer codes from algebraic codes, combining CSS, matrix-product codes, and Steane's enlargement. By leveraging three code families—Reed-Muller, hyperbolic, and affine variety codes—and their subfield-subcodes, the authors derive extensive families of stabilizer codes over various finite fields, with explicit parameters . They demonstrate numerous improvements over existing records, including binary codes of length and , and non-binary codes that meet or exceed BCH-based constructions and Gilbert-Varshamov bounds in several cases. The results provide a broad and practical toolkit for quantum error correction, with both theoretical parameters and concrete code tables applicable to quantum communication and computation.

Abstract

Stabilizer codes obtained via CSS code construction and Steane's enlargement of subfield-subcodes and matrix-product codes coming from generalized Reed-Muller, hyperbolic and affine variety codes are studied. Stabilizer codes with good quantum parameters are supplied, in particular, some binary codes of lengths 127 and 128 improve the parameters of the codes in http://www.codetables.de. Moreover, non-binary codes are presented either with parameters better than or equal to the quantum codes obtained from BCH codes by La Guardia or with lengths that can not be reached by them.

Paper Structure

This paper contains 12 sections, 19 theorems, 27 equations, 16 tables.

Table of Contents

  1. Introduction
  2. Matrix-product codes
  3. Some families of codes and their dual ones
  4. Reed-Muller codes
  5. Hyperbolic codes
  6. Affine variety codes
  7. Subfield-subcodes
  8. Quantum stabilizer codes
  9. Quantum stabilizer codes over $\mathbb{F}_2$
  10. Quantum stabilizer codes over $\mathbb{F}_3$
  11. Quantum stabilizer codes over $\mathbb{F}_q$: $q\neq 2,3$
  12. Conclusion

Key Result

Theorem 1

Let $C_1$ and $C_2$ two linear error-correcting block codes with parameters $[n,k_1,d_1]$ and $[n,k_2,d_2]$ over the field $\mathbb{F}_{q}$ and such that $C_2^{\perp} \subset C_1$, where $C_2^{\perp}$ stands for the dual code of $C_2$. Then, there exists an $[[n, k_1+k_2 -n, d]]_q$ stabilizer code which is pure to $\min \{d_1,d_2\}$, where $\mathrm{w}(c)$ denotes the weight of a word $c$.

Theorems & Definitions (28)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Theorem 3
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • proof
  • ...and 18 more