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

TL;DR

This paper explores the structure of repeated-root cyclic codes of length $2^rp^s$ over finite fields, constructing new quantum error-correcting and entanglement-assisted codes with unique parameters using CSS and Steane's methods.

Contribution

It introduces new quantum and EAQEC codes derived from repeated-root cyclic codes, including all MDS cyclic codes of specified length, with parameters different from previous work.

Findings

Constructed new QEC codes with unique parameters.

Provided all MDS cyclic codes of length $2^rp^s$.

Generated novel EAQEC codes with distinct parameters.

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.