发表机构
Chern Institute of Mathematics and LPMC, Nankai University; School of Cyberspace Security, Beijing University of Posts and Telecommunications; Department of Mathematics, University of Paris VIII; Laboratory Geometry, Analysis and Applications, LAGA, University Sorbonne Paris Nord, CNRS, UMR 7539; Telecom Paris, Polytechnic Institute of Paris; College of Science, Civil Aviation University of China（南开大学陈省身数学研究所和LPMC; 北京邮电大学网络空间安全学院; 巴黎第八大学数学系; 索邦巴黎北大学LAGA几何、分析与应用实验室; 巴黎综合理工学院巴黎电信学院; 中国民航大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出三种基于MRD码子码的最优FDRM码新构造，并利用Ferrers图适当组合的递归方法生成新最优族，解决了两个开放问题。
AI 中文摘要
本文提出了三种新的最优FDRM码构造，它们均源自最大秩距离（MRD）码的子码。第一种构造（定理5）基于系统MRD码生成矩阵的一个新族，并产生了若干此前未知的最优FDRM码。特别地，对于q≥7，它确立了参数为F=[1,2,3,4,8,8,8,8,8]的[F,7]_q FDRM码的最优性，从而解决了Zhang等人（Des. Codes Cryptogr., 87(1):107–121, 2019）提出的一个开放问题。第二种构造利用一族系统MRD码生成矩阵的结构性质，在Ferrers图F最右侧δ-2列每列至少包含n-1个点的情况下，获得新的最优FDRM码。在此基础上，我们进一步通过大幅放宽该条件来发展第三种构造：只需假设F最右侧δ-2列每列至少包含n-r个点，其中r<κ且κ=n-δ+1。此外，通过利用Ferrers图适当组合的概念，我们开发了几种递归构造，从较小的构建块生成大型FDRM码，从而产生若干新的最优族。特别地，对于具有给定参数的n×n Ferrers图F，其中一种构造在n为偶数时确立了[F,n/2-1]_q FDRM码的最优性，从而解决了Etzion等人（IEEE Trans. Inf. Theory, 62(4):1616–1630, 2016）提出的一个开放问题。
英文摘要
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 $q\geq 7$, 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 $δ-2$ 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 $δ-2$ columns of $\mathcal{F}$ 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 $n\times n$ 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).
Comments49 pages
