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

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.