Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for a given length and code size. Twisted Reed–Solomon codes over finite fields were introduced in 2017, which are generalization of Reed–Solomon codes. MDS codes can be constructed from twisted Reed–Solomon codes, and most of them are not equivalent to Reed–Solomon codes. In this paper, we give two explicit constructions of MDS twisted Reed–Solomon codes. In some cases, our constructions can obtain longer MDS codes than the constructions of previous works. Linear complementary dual (LCD) codes are linear codes that intersect with their duals trivially. LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. We also provide new constructions of LCD MDS codes from generalized twisted Reed–Solomon codes.

最大距离可分 (MDS) 代码是最佳的，因为对于给定的长度和代码大小，无法改进最小距离。2017 年引入了有限域上的 Twisted Reed-Solomon 码，它是 Reed-Solomon 码的推广。MDS 码可以由扭曲的 Reed-Solomon 码构成，其中大部分不等价于 Reed-Solomon 码。在本文中，我们给出了 MDS 扭曲 Reed-Solomon 码的两种显式构造。在某些情况下，我们的构造可以获得比以前作品的构造更长的 MDS 代码。线性互补对偶 (LCD) 码是与其对偶简单相交的线性码。LCD 代码可用于密码学。LCD 代码的这种应用重新引起了人们对构建具有大最小距离的 LCD 代码的兴趣。