Construction of MDS twisted Reed-Solomon codes and LCD MDS codes

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.