Several classes of optimal Ferrers diagram rank-metric codes

Abstract

Five constructions for Ferrers diagram rank-metric (FDRM) codes are presented. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. By introducing restricted Gabidulin codes, the second construction is presented, which unifies many known constructions for FDRM codes. The third and fourth constructions are based on two different ways of representing elements of a finite field F q m (vector representation and matrix representation). The last one is based on Ferrers diagram Kronecker products. Each of these constructions produces optimal codes with different diagrams and parameters for which no optimal construction was known before.

Bilateral multilevel construction of constant dimension codes

“…Codes attaining the bound in Lemma 2.5 are called optimal. Constructions for optimal FDRMCs can be found in [2,6,7,14,24,25,33,39]. Etzion et al presented a construction for optimal FDRMCs based on subcodes of MRD codes [6].…”

Parallel multilevel constructions for constant dimension codes

“…An FDRMC attaining the upper bound in Lemma 3.2 is called optimal. Constructions for optimal FDRMCs can be found in [2,5,6,13,17,18,24,32]. We here only quote several known constructions for late use.…”