Algebraic structures of MRD codes

Abstract

Based on results in finite geometry we prove the existence of MRD codes in $(\mathbb{F}_q)_{n,n}$ with minimum distance $n$ which are essentially different fromGabidulin codes. The construction results from algebraic structures which are closely related to those of finite fields. Some of the results may be known to experts, but to our knowledge have never been pointed out explicitly in the literature.