Entanglement-Assisted Quantum Error Correcting Codes From RS Codes and BCH Codes with Extension Degree 2

TL;DR

This paper develops explicit formulas for entanglement-assisted quantum error correcting codes derived from Reed-Solomon and BCH codes with extension degree 2, focusing on parameter computation including entanglement requirements.

Contribution

It provides a complete, explicit formula for the parameters of EAQECCs from Reed-Solomon and BCH codes with extension degree 2, including the minimum entanglement needed.

Findings

Explicit formulas for EAQECC parameters from Reed-Solomon codes.

Formulas for EAQECCs from BCH codes with extension degree 2.

Calculation of the minimum entanglement required for these codes.

Abstract

Entanglement-assisted quantum error correcting codes (EAQECCs) constructed from Reed-Solomon codes and BCH codes are considered in this work. It is provided a complete and explicit formula for the parameters of EAQECCs coming from any Reed-Solomon code, for the Hermitian metric, and from any BCH code with extension degree $2$ and consecutive cyclotomic cosets, for both the Euclidean and the Hermitian metric. The main task in this work is the computation of a completely general formula for $c$, the minimum number of required maximally entangled quantum states.