Maximal Ferrers Diagram Codes: Constructions and Genericity Considerations

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...

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Bibliographic Details
Published in:IEEE transactions on information theory 2019-10, Vol.65 (10), p.6204-6223
Main Authors: Antrobus, Jared, Gluesing-Luerssen, Heide
Format: Article
Language:English
Subjects:
Codes
Conferences
Constraining
Ferrers diagrams
Gabidulin codes
Information theory
Limiting
Measurement
Rank-metric codes
Shape
subspace codes
Task analysis
Upper bounds
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Summary: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 .
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2019.2926256