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 from Gabidulin 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.

We investigate right ideals as codes in twisted group algebras. Such codes are called twisted group codes. It turns out that many interesting codes belong to this class; for instance, the ternary extended Golay code, Hamming codes and constacyclic codes. In particular we characterize all linear codes which are twisted group codes in terms of their automorphism group.

In this paper, we propose to use a twisted dihedral group algebra for public-key cryptography. For this, we introduce a new $ 2 $-cocycle $ \alpha_{\lambda} $ to twist the dihedral group algebra. Using the ambient space $ \mathbb{F}^{\alpha_{\lambda}} D_{2n} $, we then introduce a key exchange protocol and present an analysis of its security. Moreover, we explore the properties of the resulting twisted algebra $ \mathbb{F}^{\alpha_{\lambda}}D_{2n} $, exploiting them to enhance our key exchange protocol. We also introduce a probabilistic public-key scheme derived from our key-exchange protocol and obtain a key encapsulation mechanism (KEM) by applying a well-known generic transformation to our public-key scheme. Finally, we present a proof-of-concept implementation of the resulting key encapsulation mechanism.