Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Elementary Constructions of Best Known Quantum Codes

Nuh Aydin, Trang T. T. Nguyen, Long B. Tran

TL;DR

It is shown that most of these codes can be obtained more directly from cyclic codes or their generalizations over finite fields via some version of a Gray map.

Abstract

Recently, many good quantum codes over various finite fields $F_q$ have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from cyclic codes or their generalizations over $F_q$. Unless explicit benefits are demonstrated for the indirect approach, we believe that direct and more elementary methods should be preferred.

Elementary Constructions of Best Known Quantum Codes

TL;DR

It is shown that most of these codes can be obtained more directly from cyclic codes or their generalizations over finite fields via some version of a Gray map.

Abstract

Recently, many good quantum codes over various finite fields have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from cyclic codes or their generalizations over . Unless explicit benefits are demonstrated for the indirect approach, we believe that direct and more elementary methods should be preferred.

Paper Structure

This paper contains 11 sections, 1 theorem, 13 equations, 6 tables.

Table of Contents

  1. Introduction and Motivation
  2. Basic Definitions
  3. Methods of Constructing QECCs from Classical Codes
  4. CSS Construction
  5. Lisonek-Singh Construction - Quantum Construction X
  6. Background Notation
  7. Inner products and dual codes
  8. Stabilizer codes/ Hermitian construction
  9. Computational Results
  10. Explanation of Tables
  11. Acknowledgements

Key Result

theorem 1

Let $C_1$ and $C_2$ be two linear codes over $\mathbb{F}_q$ with parameters $[n,k_1,d_1]_q$ and $[n,k_2,d_2]_q$ with $C_2^{\perp}\subseteq C_1.$ Then there exists a QECC with parameters $[[n, k_1 + k_2 - n, \min(d_1, d_2)]]_q$. In case $C_1$ is a dual-containing code, that is, $C_1^{\perp} \subseteq

Theorems & Definitions (10)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • definition 7: Generalized quasi-polycyclic code sigmaPoly
  • definition 8: 1-generator generalized quasi-polycyclic code sigmaPoly
  • definition 9
  • theorem 1: CSS construction QECC4