Near maximum distance separable (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. Quantum error-correcting codes (QECCs) play a central role in quantum information processing and can protect quantum information from various quantum noises. We present a general method for constructing QECCs over mixed alphabets through asymmetrical OAs. 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 QMDS code over mixed alphabets is proposed. By using asymmetrical OAs, we obtain many such codes.

中文翻译：

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通过正交阵列的近 MDS 和近量子 MDS 代码

近最大距离可分离（NMDS）码与有限几何中的有趣对象密切相关，并且在组合学和密码学中具有良好的应用。但NMDS代码的构建还有许多未解决的问题。在本文中，通过使用对称正交阵列（OA），我们构建了很多NMDS，米-MDS 和几乎极值的 NMDS 代码。量子纠错码（QECC）在量子信息处理中发挥着核心作用，可以保护量子信息免受各种量子噪声的影响。我们提出了一种通过非对称 OA 在混合字母表上构建 QECC 的通用方法。由于迄今为止所有文献中尚未发现维数等于1的混合字母表上的量子最大距离可分离(QMDS)码，因此提出了混合字母表上的近QMDS码的定义。通过使用非对称OA，我们获得了许多这样的代码。