# Gabidulin Codes with Support Constrained Generator Matrices

## TL;DR

This paper establishes conditions for constructing Gabidulin codes with generator matrices under support constraints, enabling their application in network coding and distributed systems while characterizing maximum achievable rank distance.

## Contribution

It provides necessary and sufficient support conditions for Gabidulin codes and characterizes maximum rank distance under support constraints, extending prior MDS code results.

## Key findings

- Support conditions for Gabidulin codes are established.
- Maximum rank distance under support constraints is characterized.
- Subcodes of Gabidulin codes can achieve optimal rank distance.

## Abstract

Gabidulin codes are the first general construction of linear codes that are maximum rank distant (MRD). They have found applications in linear network coding, for example, when the transmitter and receiver are oblivious to the inner workings and topology of the network (the so-called incoherent regime). The reason is that Gabidulin codes can be used to map information to linear subspaces, which in the absence of errors cannot be altered by linear operations, and in the presence of errors can be corrected if the subspace is perturbed by a small rank. Furthermore, in distributed coding and distributed systems, one is led to the design of error correcting codes whose generator matrix must satisfy a given support constraint. In this paper, we give necessary and sufficient conditions on the support of the generator matrix that guarantees the existence of Gabidulin codes and general MRD codes. When the rate of the code is not very high, this is achieved with the same field size necessary for Gabidulin codes with no support constraint. When these conditions are not satisfied, we characterize the largest possible rank distance under the support constraints and show that they can be achieved by subcodes of Gabidulin codes. The necessary and sufficient conditions are identical to those that appear for MDS codes which were recently proven by Yildiz et al. and Lovett in the context of settling the GM-MDS conjecture.