m-QMDS codes over mixed alphabets via orthogonal arrays (2406.10629v1)

Abstract: The construction of quantum error-correcting codes (QECCs) with good parameters is a hot topic in the area of quantum information and quantum computing. Quantum maximum distance separable (QMDS) codes are optimal because the minimum distance cannot be improved for a given length and code size. The QMDS codes over mixed alphabets are rarely known even if the existence and construction of QECCs over mixed alphabets with minimum distance more than or equal to three are still an open question. In this paper, we define an $m$-QMDS code over mixed alphabets, which is a generalization of QMDS codes. We establish a relation between $m$-QMDS codes over mixed alphabets and asymmetrical orthogonal arrays (OAs) with orthogonal partitions. Using this relation, we propose a general method to construct $m$-QMDS codes. As applications of this method, numerous infinite families of $m$-QMDS codes over mixed alphabets can be constructed explicitly. Compared with existing codes, the constructed codes have more flexibility in the choice of parameters, such as the alphabet sizes, length and dimension of the encoding state.