A geometric invariant of linear rank-metric codes
Valentina Astore, Martino Borello, Marco Calderini, Flavio Salizzoni
TL;DR
The paper introduces a geometric invariant for linear rank-metric codes by extending Schur-product ideas from the Hamming metric to rank-metric codes through the extended Hamming code and associated linear sets. It defines the $\mathbb{F}_q$-dimension (Hilbert) sequence $h_i(\mathcal{C})$ from the Hamming-associated code $\mathcal{C}^{\rm H}$ and proves a sharp formula $h_{q+1}(\mathcal{C})=\binom{k+q}{q+1}-\binom{k-r}{2}$ with $r=\operatorname{rk}(X^{[1]}-X)$; this yields a robust discriminant between Gabidulin codes (where $r=1$) and random codes via a Schwartz–Zippel argument. The work further connects higher terms $h_{q^s+1}(\mathcal{C})$ to the intersection $\mathcal{F}_s\cap \mathcal{I}(L_{\mathcal{U}_G})$ of polynomial spaces with the vanishing ideal of the linear set, proving that $\operatorname{rk}(X^{[s]}-X)=r$ iff $\dim \mathcal{F}_s\cap \mathcal{I}(L_{\mathcal{U}_G})=\binom{k-r}{2}$. By tying Hilbert-function-based invariants to vanishing-ideal geometry of linear sets, the paper provides both algebraic and geometric tools for recognizing structured rank-metric codes and establishes bounds on zeros of associated forms, with potential cryptographic and code-classification implications.
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.