线性秩度量码的一个几何不变量
A geometric invariant of linear rank-metric codes
AI总结：
本文受Schur积启发，为线性秩度量码提出几何不变量，通过Schur幂维数序列区分Gabidulin码与随机码。
摘要：
秩度量码因其理论和实践意义而成为编码理论的核心课题，其应用包括网络编码、分布式存储、交叉纠错和后量子密码学。近期研究侧重于构造具有不同代数结构的新型秩度量码族，强调了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Schur积的启发，为线性秩度量码引入了一个新颖的几何不变量。通过考察与线性码相关联的扩展汉明码的Schur幂的维数序列，我们证明了该不变量能够区分Gabidulin码与随机码。从几何角度来看，此方法研究了与秩度量码对应的线性集的消失理想。
英文摘要：
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.
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