Asymmetric entanglement-assisted quantum error-correcting MDS codes of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$

Abstract

In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$ . We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. If we fix the physical qubits and the logical qubits of EAQEC codes, many of them have larger quantum distances than the well known EAQEC codes in the literature. Moreover, some of our EAQEC codes achieve larger quantum distances while requiring fewer or the same number of entangled bits compared with existing EAQEC codes when the numbers of physical qubits and logical qubits are fixed. We construct asymmetric entanglement-assisted quantum error-correcting (AEAQEC) MDS codes of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$ . When the physical qubits of the AEAQEC codes is fixed, many of our AEAQEC codes have a significantly larger number of quantum distances compared to all previously constructed AEAQEC codes. We also provide a version of Singleton bound for all AEAQEC codes.