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New QEC codes and EAQEC codes from repeated-root cyclic codes of length $2^rp^s$

Lanqiang Li, Ziwen Cao, Tingting Wu, Li Liu

TL;DR

This study begins its exploration by delving into the intricate structure of all repeated-root cyclic codes and their duals with a length of $2^rp^s$ over the finite field $\mathbb{F}_{p^m}$ through the utilization of CSS and Steane's constructions.

Abstract

Let $p$ be an odd prime and $r,s,m$ be positive integers. In this study, we initiate our exploration by delving into the intricate structure of all repeated-root cyclic codes and their duals with a length of $2^rp^s$ over the finite field $\mathbb{F}_{p^m}$. Through the utilization of CSS and Steane's constructions, a series of new quantum error-correcting (QEC) codes are constructed with parameters distinct from all previous constructions. Furthermore, we provide all maximum distance separable (MDS) cyclic codes of length $2^rp^s$, which are further utilized in the construction of QEC MDS codes. Finally, we introduce a significant number of novel entanglement-assisted quantum error-correcting (EAQEC) codes derived from these repeated-root cyclic codes. Notably, these newly constructed codes exhibit parameters distinct from those of previously known constructions.

New QEC codes and EAQEC codes from repeated-root cyclic codes of length $2^rp^s$

TL;DR

This study begins its exploration by delving into the intricate structure of all repeated-root cyclic codes and their duals with a length of over the finite field through the utilization of CSS and Steane's constructions.

Abstract

Let be an odd prime and be positive integers. In this study, we initiate our exploration by delving into the intricate structure of all repeated-root cyclic codes and their duals with a length of over the finite field . Through the utilization of CSS and Steane's constructions, a series of new quantum error-correcting (QEC) codes are constructed with parameters distinct from all previous constructions. Furthermore, we provide all maximum distance separable (MDS) cyclic codes of length , which are further utilized in the construction of QEC MDS codes. Finally, we introduce a significant number of novel entanglement-assisted quantum error-correcting (EAQEC) codes derived from these repeated-root cyclic codes. Notably, these newly constructed codes exhibit parameters distinct from those of previously known constructions.
Paper Structure (7 sections, 25 theorems, 47 equations, 8 tables)

This paper contains 7 sections, 25 theorems, 47 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cyclic codes of length $2^rp^s$ and their duals
  4. QEC codes of length $2^rp^s$
  5. QEC MDS codes of length $2^rp^s$
  6. EAQEC codes of length $2^rp^s$
  7. Conclusion

Key Result

Theorem 1

For any $[[n,k,d]]_q$ QEC code, its parameters must satisfy: $k\leq n-2d+2$. The code is referred to as a QEC MDS code if the equality holds.

Theorems & Definitions (55)

  • Definition 1
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • Lemma 6
  • Proposition 7
  • ...and 45 more