Code distances of matrix rank-metric codes under Add-and-Remove transformations

2026-10-02

AI Summary

This study investigates the evolution of subcode distances in matrix rank-metric codes under the Add-and-Remove construction. By integrating Delsarte duality theory, linear algebra, and coding-theoretic methods, we derive subcode distance inequalities and explicit bounds for arbitrary codes as well as Gabidulin codes. Our analysis reveals the intrinsic relationship between the minimum distance of the newly constructed code and the added space, alongside a parameter-exchange property arising from duality. Furthermore, this work provides a rigorous proof of the distance lower bound for the MIRANDA signature scheme, thereby solidifying its theoretical security foundation.

Abstract

For a matrix rank-metric code $\mathcal{C}$ and an integer $i$, the $i$-th subcode distance of $\mathcal{C}$ is the largest minimum rank distance of an $i$-dimensional subcode of $\mathcal{C}$. We study how subcode distances change under the Add-and-Remove construction, in which a subcode of codimension $\ell_s$ is kept and a space of dimension $\ell_a$ is added; this construction is used, with Gabidulin codes, in the $\mathsf{MIRANDA}$ signature scheme. For arbitrary matrix codes, we give inequalities between the subcode distances of the original and of the new code, and we show that Delsarte duality interchanges $\ell_s$ and $\ell_a$. For matrix Gabidulin codes, whose subcode distances are given by an explicit formula, we show that, if $\ell_s<\max\{m,n\}$, then within each block of indices $(a-1)\max\{m,n\}+1,\ldots,a\max\{m,n\}$ not exceeding $\dim_{\mathbb{F}_q}(\mathcal{C}_s)$, all but the last $\ell_s$ subcode distances are the same as for the Gabidulin code, and the remaining ones decrease by at most one; a small example shows that such a decrease can occur. We also express the minimum distance of the new code in terms of the rank distance between the added space and the common subcode. For $\mathsf{MIRANDA}$, the lower bound on the minimum distance of the dual public code used by the known distinguisher when $\ell_s=0$ is a special case of these results; for the proposed parameters, where $\ell_s>0$, the same bound holds for a subcode of codimension $\ell_s$ of the dual public code.

Problem

• matrix rank-metric codes
• subcode distances
• Add-and-Remove transformations
• Gabidulin codes
• MIRANDA signature scheme

Innovation

• Rank-metric codes
• Subcode distance
• Add-and-Remove construction
• Gabidulin codes
• MIRANDA signature scheme