Optimal Ferrers Diagram Rank-Metric Codes: New Constructions, Diagram Combinations, and Applications to Constant-Dimension Subspace Codes

Abstract

In this paper, we present three new constructions of optimal FDRM codes, all derived from subcodes of maximum rank-distance (MRD) codes. The first construction (Theorem~\ref{theo5}) is based on a new family of generator matrices for systematic MRD codes and yields several previously unknown optimal FDRM codes. In particular, for $q\ge 7$, it establishes the optimality of $[\mathcal{F},7{]}_q$ FDRM codes with $\mathcal{F}=[1,2,3,4,8,8,8,8,8]$, thereby resolving an open problem posed by Zhang \emph{et al.} (Des. Codes Cryptogr., 87(1):107--121, 2019). Our second construction exploits structural properties of generator matrices of a family of systematic MRD codes to obtain new optimal FDRM codes whenever each of the rightmost $\delta -2$ columns of the Ferrers diagram $\mathcal{F}$ contains at least $n-1$ dots. Building upon this approach, we further develop a third construction by substantially relaxing this requirement: it is sufficient to assume that each of the rightmost $\delta -2$ columns of $\mathcal{F}$ contains at least $n-r$ dots, where $r<\kappa$ and $\kappa =n-\delta +1$. Furthermore, by exploiting the notion of proper combinations of Ferrers diagrams, we develop several recursive constructions that produce large FDRM codes from smaller building blocks, yielding a number of new optimal families. In particular, for an $n\times n$ Ferrers diagram $\mathcal{F}$ with prescribed parameters, one of these constructions establishes the optimality of $[\mathcal{F},\frac{n}{2}-1{]}_q$ FDRM codes whenever $n$ is even, thereby settling an open problem posed by Etzion \emph{et al.} (IEEE Trans. Inf. Theory, 62(4):1616--1630, 2016).

49 pages