Year: 2025
Type: article
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.
DOI: 10.1137/24m1713910
Institutions: Institut national de recherche en sciences et technologies du numérique, École Polytechnique, Centre Inria de Saclay, Laboratoire d'Informatique de l'École Polytechnique, Institut Polytechnique de Paris, GRACE: Geometry, arithmetic, algorithms, codes and encryption, Centre National de la Recherche Scientifique, Université Sorbonne Paris Nord, Laboratoire Analyse, Géométrie et Applications, Université Paris 8, University of Trento, Max Planck Institute for Mathematics in the Sciences, Max Planck Institute for Mathematics
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4405029994