Rank distribution of Delsarte codes
Javier de la Cruz, Elisa Gorla, Hiram H. Lopez, Alberto Ravagnani
TL;DR
The paper defines rank defect $\mathrm{Rdef}(\mathcal{C}) = n - \left\lceil t/m \right\rceil - d + 1$ for Delsarte rank-metric codes and classifies codes as MRD, QMRD, or dually QMRD. It establishes MacWilliams identities for rank distributions and proves that, for MRD or dually QMRD codes, the entire rank distribution is determined by the parameters $n$, $m$, $t$, and $d$ (with an explicit closed form) and, more generally, that the distribution is determined by these parameters plus $A_d(\mathcal{C}),\ldots,A_{n-d^{\perp}}(\mathcal{C})$; codes with small rank defect satisfy symmetry in the minimum-rank counts between a code and its dual when $d+d^{\perp}=n$ and $m|t$. The results specialize to Gabidulin codes, where MRD and AMRD notions yield parallel distribution formulas and duality properties, connecting rank-metric theory to classical MRD/Gabidulin results. Overall, the work clarifies how rank distribution is governed by structural parameters and a few low-weight terms, with practical implications for code design in rank-metric settings and Gabidulin-based applications.
Abstract
In analogy with the Singleton defect for classical codes, we propose a definition of rank defect for Delsarte rank-metric codes. We characterize codes whose rank defect and dual rank defect are both zero, and prove that the rank distribution of such codes is determined by their parameters. This extends a result by Delsarte on the rank distribution of MRD codes. In the general case of codes of positive defect, we show that the rank distribution is determined by the parameters of the code, together the number of codewords of small rank. Moreover, we prove that if the rank defect of a code and its dual are both one, and the dimension satisfies a divisibility condition, then the number of minimum-rank codewords and dual minimum-rank codewords is the same. Finally, we discuss how our results specialize to Gabidulin codes.