Rank-metric codes have been a central topic in coding theory due to their theoretical and practical significance, with applications in network coding, distributed storage, crisscross error correction, and post-quantum cryptography. Recent research has focused on constructing new families of rank-metric codes with distinct algebraic structures, emphasizing the importance of invariants for distinguishing these codes from known families and from random ones. In this paper, we introduce a novel geometric invariant for linear rank-metric codes, inspired by the Schur product used in the Hamming metric. By examining the sequence of dimensions of Schur powers of the extended Hamming code associated with a linear code, we demonstrate its ability to differentiate Gabidulin codes from random ones. From a geometric perspective, this approach investigates the vanishing ideal of the linear set corresponding to the rank-metric code.

翻译：秩度量码因其理论和实际意义已成为编码理论的核心课题，在**网络编码、分布式存储、交叉纠错和后量子密码学**等领域具有重要应用。近期研究聚焦于构建具有不同代数结构的新秩度量码族，强调不变量在区分这些码与已知码族及随机码方面的重要性。本文受汉明度量中**舒尔积**的启发，为线性秩度量码引入了一种新的几何不变量。通过考察与线性码关联的扩展汉明码的舒尔幂维数序列，我们证明了该不变量能够区分**Gabidulin码**与随机码。从几何视角看，此方法研究了对应于秩度量码的线性集的消失理想。