New LCD MDS codes of non-Reed-Solomon type

TL;DR

This paper introduces new classes of Euclidean and Hermitian LCD MDS codes that are not based on Reed-Solomon codes, expanding the variety of algebraic structures available for practical applications.

Contribution

It constructs the first known LCD MDS codes of non-Reed-Solomon type, not monomially equivalent to existing Reed-Solomon-based codes, using novel algebraic methods.

Findings

First construction of non-Reed-Solomon LCD MDS codes

Codes are not monomially equivalent to Reed-Solomon codes

Expands the diversity of algebraic structures for LCD MDS codes

Abstract

Both linear complementary dual (LCD) codes and maximum distance separable (MDS) codes have good algebraic structures, and they have interesting practical applications such as communication systems, data storage, quantum codes, and so on. So far, most of LCD MDS codes have been constructed by employing generalized Reed-Solomon codes. In this paper we construct some classes of new Euclidean LCD MDS codes and Hermitian LCD MDS codes which are not monomially equivalent to Reed-Solomon codes, called LCD MDS codes of non-Reed- Solomon type. Our method is based on the constructions of Beelen et al. (2017) and Roth and Lempel (1989). To the best of our knowledge, this is the first paper on the construction of LCD MDS codes of non-Reed-Solomon type; any LCD MDS code of non- Reed-Solomon type constructed by our method is not monomially equivalent to any LCD code constructed by the method of Carlet et al. (2018).