We determine the proportion of linear [3×3;3]-MRD codes over Fq within the space of all 3-dimensional 3×3-rank-metric codes over the same field. This shows that for these parameters linear MRD codes are sparse in the sense that the proportion tends to 0 as q→∞. This is so far the only parameter case for which MRD codes are known to be sparse. The computation is accomplished by reducing the space of all such rank-metric codes to a space of specific bases and subsequently making use of a result by Menichetti (1973) on 3-dimensional semifields.
|Number of pages||24|
|Journal||Linear Algebra and Its Applications|
|State||Published - Jul 1 2020|
Bibliographical noteFunding Information:
HGL was partially supported by the grant #422479 from the Simons Foundation.
© 2020 Elsevier Inc.
- MRD codes
- Rank-metric codes
ASJC Scopus subject areas
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics