This paper investigates the construction of rank-metric codes with specified Ferrers diagram shapes. These codes play a role in the multilevel construction for subspace codes. A conjecture from 2009 provides an upper bound for the dimension of a rank-metric code with given specified Ferrers diagram shape and rank distance. While the conjecture in its generality is wide open, several cases have been established in the literature. This paper contributes further cases of Ferrers diagrams and ranks for which the conjecture holds true. In addition, the proportion of maximal Ferrers diagram codes within the space of all rank-metric codes with the same shape and dimension is investigated. Special attention is being paid to MRD codes. It is shown that for growing field size the limiting proportion depends highly on the Ferrers diagram. For instance, for [m\times 2] -MRD codes with rank 2 this limiting proportion is close to 1/e.
|Number of pages||20|
|Journal||IEEE Transactions on Information Theory|
|State||Published - Oct 2019|
Bibliographical noteFunding Information:
Manuscript received May 4, 2018; revised April 26, 2019; accepted June 26, 2019. Date of publication July 1, 2019; date of current version September 13, 2019. H. Gluesing-Luerssen was supported in part by the Simons Foundation under Grant 422479. This paper was presented in part at the 2018 AMS Sectional Meeting, March, and in part at the 2018 Dagstuhl Workshop on Algebraic Coding Theory for Networks, Storage, and Security.
© 1963-2012 IEEE.
- Ferrers diagrams
- Gabidulin codes
- Rank-metric codes
- subspace codes
ASJC Scopus subject areas
- Information Systems
- Computer Science Applications
- Library and Information Sciences