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Near MDS (NMDS) codes are closely related to interesting objects in finitegeometry and have nice applications in combinatorics and cryptography. Butthere are many unsolved problems about construction of NMDS codes. In thispaper, by using symmetrical orthogonal arrays (OAs), we construct a lot ofNMDS, $m$-MDS and almost extremal NMDS codes. We establish a relation betweenasymmetrical OAs and quantum error correcting codes (QECCs) over mixedalphabets. Since quantum maximum distance separable (QMDS) codes over mixedalphabets with the dimension equal to one have not been found in all theliterature so far, the definition of a near quantum maximum distance separable(NQMDS) code over mixed alphabets is proposed. By using asymmetrical OAs, weobtain many such codes.

Near MDS and near quantum MDS codes via orthogonal arrays