Yes, the techniques employed in this paper can be extended to investigate perfect codes and covering properties in other restricted matrix spaces, such as alternating matrices or symmetric matrices. The foundational concepts of sphere-packing bounds and covering density are applicable across various types of matrix spaces, as they rely on the rank metric and the properties of the matrices involved. For instance, the methods used to establish bounds on the size of spheres and balls in Hermitian rank-metric codes can similarly be applied to alternating matrices, where the rank metric is also relevant.
Moreover, the analysis of perfect codes, which involves demonstrating the absence of non-trivial perfect codes through the application of the sphere-packing bound, can be adapted to other restricted matrix spaces. The underlying principles of rank distance and the associated bounds can provide insights into the existence and structure of perfect codes in these contexts. By leveraging the established results and methodologies from the study of Hermitian rank-metric codes, researchers can explore the unique properties and potential limitations of perfect codes in alternating and symmetric matrix spaces.