Systematic encoders for generalized Gabidulin codes and the $q$-analogue of Cauchy matrices

We characterize the generator matrix in standard form of generalized Gabidulin codes. The parametrization we get for the non-systematic part of this matrix coincides with the $q$-analogue of generalized Cauchy matrices, leading to the definition of $q$-Cauchy matrices. These matrices can be represented very conveniently and their representation allows to define new interesting subfamilies of generalized Gabidulin codes whose generator matrix is a structured matrix. In particular, as an application, we construct Gabidulin codes whose generator matrix is the concatenation of an identity block and a Toeplitz/Hankel matrix. In addition, our results allow to give a new efficient criterion to verify whether a rank metric code of dimension $k$ and length $n$ is a generalized Gabidulin code. This criterion is only based on the computation of the rank of one matrix and on the verification of the linear independence of two sets of elements and it requires $\mathcal O(m\cdot F(k,n))$ field operations, where $F(k,n)$ is the cost of computing the reduced row echelon form of a $k \times n$ matrix. Moreover, we also provide a characterization of the generator matrix in standard form of general MRD codes.