A geometric invariant of linear rank-metric codes

Abstract

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.

21 pages, 2 figures

References in corpus (9)

• A Rank-Metric Approach to Error Control in Random Network Coding
• Generalized Twisted Gabidulin Codes
• Squares of Random Linear Codes
• New Criteria for MRD and Gabidulin Codes and some Rank-Metric Code Constructions
• On the Genericity of Maximum Rank Distance and Gabidulin Codes
• New Semifields and new MRD Codes from Skew Polynomial Rings
• Critical pairs for the Product Singleton Bound
• Identifiers for MRD-codes
• Equivalence and Characterizations of Linear Rank-Metric Codes Based on Invariants