# MRD Codes: Constructions and Connections

## TL;DR

This chapter surveys the constructions, applications, and open problems related to Maximum Rank Distance (MRD) codes, which are crucial in network coding and connect to various mathematical fields.

## Contribution

It provides a comprehensive overview of known MRD code constructions, their applications, and highlights open problems in the area.

## Key findings

- MRD codes are optimal rank-metric codes with maximum size for given parameters.
- Connections exist between MRD codes and semifields, finite geometry, and cryptography.
- Recent interest is driven by applications in network coding.

## Abstract

This preprint is of a chapter to appear in {\it Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications. Radon Series on Computational and Applied Mathematics}, K.-U. Schmidt and A. Winterhof (eds.). Rank-metric codes are codes consisting of matrices with entries in a finite field, with the distance between two matrices being the rank of their difference. Codes with maximum size for a fixed minimum distance are called Maximum Rank Distance (MRD) codes. Such codes were constructed and studied independently by Delsarte (1978), Gabidulin (1985), Roth (1991), and Cooperstein (1998). Rank-metric codes have seen renewed interest in recent years due to their applications in random linear network coding. MRD codes also have interesting connections to other topics such as semifields (finite nonassociative division algebras), finite geometry, linearized polynomials, and cryptography. In this chapter we will survey the known constructions and applications of MRD codes, and present some open problems.