A geometric invariant of linear rank-metric codes

2024-11-28

SIAM Journal on applied algebra and geometry

AI Summary

This work addresses the indistinguishability problem between Gabidulin codes and random linear rank-metric codes. We introduce the first geometric invariant tailored to the rank metric. Inspired by the Schur product dimension sequence in the Hamming metric, we generalize the Schur product to the rank-metric setting and establish a geometric correspondence between linear rank-metric codes and the vanishing ideals of their associated linear sets. Integrating projective geometry with algebraic techniques, we construct an invariant characterized by the sequence of dimensions of Schur powers. This invariant efficiently captures structural properties unique to Gabidulin codes and distinguishes them from random linear rank-metric codes in polynomial time. Our results provide a novel theoretical framework and computational tool for classification, structural identification, and cryptographic security analysis of rank-metric codes.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Introducing a novel geometric invariant for linear rank-metric codes
• Distinguishing structured rank-metric codes from random codes
• Investigating vanishing ideals of linear sets in rank-metric codes

Innovation

• Introduces geometric invariant for linear rank-metric codes
• Uses Schur product dimensions from extended Hamming code
• Analyzes vanishing ideals of corresponding linear sets