Constructing MRD codes by switching

Maximum rank‐distance (MRD) codes are (not necessarily linear) maximum codes in the rank‐distance metric space on m $m$‐by‐n $n$ matrices over a finite field Fq ${{\mathbb{F}}}_{q}$. They are diameter perfect and have the cardinality qm(n−d+1) ${q}^{m(n-d+1)}$ if m≥n $m\ge n$. We define switching in...

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Bibliographic Details
Published inJournal of combinatorial designs Vol. 32; no. 5; pp. 219 - 237
Main Authors Shi, Minjia, Krotov, Denis S., Özbudak, Ferruh
Format Journal Article
LanguageEnglish
Published Hoboken Wiley Subscription Services, Inc 01.05.2024
Subjects
bilinear forms graph
Fields (mathematics)
Metric space
MRD codes
Parameters
rank distance
Switching
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Summary:Maximum rank‐distance (MRD) codes are (not necessarily linear) maximum codes in the rank‐distance metric space on m $m$‐by‐n $n$ matrices over a finite field Fq ${{\mathbb{F}}}_{q}$. They are diameter perfect and have the cardinality qm(n−d+1) ${q}^{m(n-d+1)}$ if m≥n $m\ge n$. We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting switching, such as punctured twisted Gabidulin codes and direct‐product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in m $m$ if the other parameters (n,q $n,\,q$, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes.
ISSN:1063-8539
1520-6610
DOI:10.1002/jcd.21931