A geometric invariant of linear rank-metric codes

TL;DR

This paper introduces a new geometric invariant for linear rank-metric codes, inspired by the Schur product, which helps distinguish Gabidulin codes from random codes by analyzing their algebraic and geometric properties.

Contribution

The paper presents a novel geometric invariant based on Schur powers and vanishing ideals, providing a new tool for classifying and differentiating rank-metric codes.

Findings

The invariant distinguishes Gabidulin codes from random codes.

It is based on the sequence of dimensions of Schur powers.

The approach links algebraic and geometric perspectives of 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.