Constructing MRD codes by switching

TL;DR

This paper introduces a switching technique for constructing a vast new class of MRD codes with enhanced parameters, including codes with varying affine ranks and aperiodic structures, expanding the known landscape of rank-metric codes.

Contribution

It develops a novel switching method for MRD codes, enabling the creation of many new codes with larger size and diverse properties, such as different affine ranks and aperiodicity.

Findings

Constructed a large class of MRD codes with doubly exponential size growth.

Developed constructions for MRD codes with varying affine ranks.

Created aperiodic MRD codes with new structural properties.

Abstract

MRD codes are maximum codes in the rank-distance metric space on $m$-by-$n$ matrices over the finite field of order $q$. They are diameter perfect and have the cardinality $q^{m(n-d+1)}$ if $m\ge n$. We define switching in MRD codes as replacing special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting such switching, including punctured twisted Gabidulin codes and direct-product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in $m$ if the other parameters ($n$, $q$, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes. Keywords: MRD codes, rank distance, bilinear forms graph, switching, diameter perfect codes