Near MDS and near quantum MDS codes via orthogonal arrays

TL;DR

This paper introduces new constructions of near MDS and near quantum MDS codes using orthogonal arrays, addressing open problems in code construction and establishing links between combinatorial designs and quantum error correction.

Contribution

It presents novel methods to construct NMDS, m-MDS, and NQMDS codes via orthogonal arrays, and explores their relation to quantum error correcting codes over mixed alphabets.

Findings

Constructed numerous NMDS, m-MDS, and almost extremal NMDS codes.

Established a relation between asymmetrical orthogonal arrays and quantum error correcting codes.

Proposed the concept of near quantum MDS codes over mixed alphabets.

Abstract

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