Rank-metric codes, defined as sets of matrices over a finite field with the rank distance, have gained significant attention due to their applications in network coding and connections to diverse mathematical areas. Initially studied by Delsarte in 1978 and later rediscovered by Gabidulin, these codes have become a central topic in coding theory. This paper surveys the development and mathematical foundations, in particular, regarding bounds and constructions of rank-metric codes, emphasizing their extension beyond finite fields to more general settings. We examine Singleton-like bounds on code parameters, demonstrating their sharpness in finite field cases and contrasting this with contexts where the bounds are not tight. Furthermore, we discuss constructions of Maximum Rank Distance (MRD) codes over fields with cyclic Galois extensions and the relationship between linear rank-metric codes with systems and evasive subspaces. The paper also reviews results for algebraically closed fields and real numbers, previously appearing in the context of topology and measure theory. We conclude by proposing future research directions, including conjectures on MRD code existence and the exploration of rank-metric codes over various field extensions.

翻译：秩度量码定义为有限域上具有秩距离的矩阵集合，由于其在网络编码中的应用以及与多个数学领域的联系而受到广泛关注。这类码最初由Delsarte于1978年研究，后由Gabidulin重新发现，现已成为编码理论的核心课题。本文综述了秩度量码的发展历程与数学基础，特别聚焦于码的界与构造方法，并重点探讨其从有限域向更一般数域的扩展。我们研究了码参数的类Singleton界，证明了其在有限域情形下的紧性，并与界不紧的情形进行对比。此外，我们讨论了具有循环伽罗瓦扩张的域上最大秩距离（MRD）码的构造方法，以及线性秩度量码与系统及规避子空间之间的关系。本文还回顾了代数闭域和实数域上的相关结果，这些结果先前曾出现在拓扑学和测度论的语境中。最后，我们提出了未来的研究方向，包括关于MRD码存在性的猜想以及对各类域扩张上秩度量码的探索。