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≥ 7, it establishes the optimality of [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 δ−2 columns of the Ferrers diagram 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 δ−2 columns of ℱ contains at least n−r dots, where r<κ and κ=n−δ+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× n Ferrers diagram ℱ with prescribed parameters, one of these constructions establishes the optimality of [ℱ,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).
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BibTeX
@misc{fu2026optimal,
title = {{Optimal Ferrers Diagram Rank-Metric Codes: New Constructions, Diagram Combinations, and Applications to Constant-Dimension Subspace Codes}},
author = {Fang-Wei Fu and Xuan Gao and Sihem Mesnager and Gang Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24239},
eprint = {2609.24239},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24239},
}
RIS
TY - UNPB
TI - Optimal Ferrers Diagram Rank-Metric Codes: New Constructions, Diagram Combinations, and Applications to Constant-Dimension Subspace Codes
AU - Fang-Wei Fu
AU - Xuan Gao
AU - Sihem Mesnager
AU - Gang Wang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24239
UR - https://arxiv.org/abs/2609.24239
AB - 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≥ 7, it establishes the optimality of [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 δ−2 columns of the Ferrers diagram 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 δ−2 columns of ℱ contains at least n−r dots, where r<κ and κ=n−δ+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× n Ferrers diagram ℱ with prescribed parameters, one of these constructions establishes the optimality of [ℱ,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).
ER -