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Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes

Martianus Frederic Ezerman, Markus Grassl, San Ling, Ferruh Özbudak, Buket Özkaya

TL;DR

This work generalizes Quantum Construction X by incorporating nearly self-orthogonal quasi-twisted codes under Hermitian, symplectic, and trace-symplectic inner products to build quantum stabilizer codes. It provides constructive proofs and refined lower bounds on the quantum minimum distance $d$, aided by hull-design techniques and Gram-Schmidt-type orthonormalization, enabling systematic hull control. A comprehensive randomized search, plus propagation rules, yields numerous record-breaking $q$-ary quantum codes, expanding the catalog of practical stabilizer codes and updating online databases. The results have substantial significance for quantum error correction, offering new coding-theoretic pathways and explicit parameters that advance quantum coding theory and its applications in quantum communication and computation.

Abstract

Quasi-twisted codes are used here as the classical ingredients in the so-called Construction X for quantum error-control codes. The construction utilizes nearly self-orthogonal codes to design quantum stabilizer codes. We expand the choices of the inner product to also cover the symplectic and trace-symplectic inner products, in addition to the original Hermitian one. A refined lower bound on the minimum distance of the resulting quantum codes is established and illustrated. We report numerous record breaking quantum codes from our randomized search for inclusion in the updated online database.

Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes

TL;DR

This work generalizes Quantum Construction X by incorporating nearly self-orthogonal quasi-twisted codes under Hermitian, symplectic, and trace-symplectic inner products to build quantum stabilizer codes. It provides constructive proofs and refined lower bounds on the quantum minimum distance , aided by hull-design techniques and Gram-Schmidt-type orthonormalization, enabling systematic hull control. A comprehensive randomized search, plus propagation rules, yields numerous record-breaking -ary quantum codes, expanding the catalog of practical stabilizer codes and updating online databases. The results have substantial significance for quantum error correction, offering new coding-theoretic pathways and explicit parameters that advance quantum coding theory and its applications in quantum communication and computation.

Abstract

Quasi-twisted codes are used here as the classical ingredients in the so-called Construction X for quantum error-control codes. The construction utilizes nearly self-orthogonal codes to design quantum stabilizer codes. We expand the choices of the inner product to also cover the symplectic and trace-symplectic inner products, in addition to the original Hermitian one. A refined lower bound on the minimum distance of the resulting quantum codes is established and illustrated. We report numerous record breaking quantum codes from our randomized search for inclusion in the updated online database.
Paper Structure (19 sections, 19 theorems, 96 equations, 1 figure, 8 tables)

This paper contains 19 sections, 19 theorems, 96 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Inner Products and Dual Codes
  4. Quantum Codes
  5. Generalized Lisoněk-Singh Construction
  6. The Hermitian Case
  7. The Symplectic Case
  8. The Trace-Symplectic Case
  9. Quasi-Twisted Codes with Designed Hermitian Hulls
  10. Quasi-Twisted Codes with Designed Symplectic Hulls
  11. Quasi-Cyclic and Quasi-Negacyclic Codes
  12. $(\lambda_1,\lambda_2)$-Quasi-Twisted Codes
  13. New Quantum Codes
  14. General Remarks
  15. Comments on the Randomized Search
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $C\subseteq{\mathbb F}_q^{2n}$ be a trace-symplectic self-orthogonal additive code, i. e., $C\subseteq C^{\perp_{\rm T}}$, of size $|C|=q^n/K$. Then there exists an $(\!(n,K,d(\mathcal{Q}))\!)_q$ quantum stabilizer code $\mathcal{Q}$ with For $C\ne C^{\perp_{\rm T}}$, i. e., $K>1$, the code is called impure whenever $d(\mathcal{Q})>\mathop{\rm swt}(C^{\perp_{\rm T}})$; otherwise the code is c

Figures (1)

  • Figure 1: Illustration of the different spaces related to the matrix in \ref{['eq:genmat_H']} as well as the matrices in \ref{['eq:genmat_symp']} and \ref{['eq:genmat_CSS']}. The symbol ${}^\perp$ denotes the corresponding notion of duality, and the matrices $B$, $M$, and $A$ might be formed by several blocks.

Theorems & Definitions (27)

  • Proposition 2.1: trace-symplectic construction
  • Proposition 2.2: symplectic construction
  • Proposition 2.3: CSS construction
  • Proposition 2.4: Hermitian construction
  • Theorem 3.1
  • Example 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • Theorem 3.6
  • ...and 17 more