Codes and Designs Related to Lifted MRD Codes

TL;DR

This paper explores the structure of lifted MRD codes, representing them as transversal designs, and uses this to derive bounds and construct large constant dimension codes, also introducing related linear codes.

Contribution

It reveals that lifted MRD codes form transversal designs and uses this to establish bounds and construct optimal codes, also deriving new linear codes in Hamming space.

Findings

Lifted MRD codes can be represented as transversal designs.

Bounds on the sizes of constant dimension codes containing lifted MRD codes are established.

New large codes and linear codes in Hamming space are constructed based on these designs.

Abstract

Lifted maximum rank distance (MRD) codes, which are constant dimension codes, are considered. It is shown that a lifted MRD code can be represented in such a way that it forms a block design known as a transversal design. A slightly different representation of this design makes it similar to a $q-$analog of a transversal design. The structure of these designs is used to obtain upper bounds on the sizes of constant dimension codes which contain a lifted MRD code. Codes which attain these bounds are constructed. These codes are the largest known codes for the given parameters. These transversal designs can be also used to derive a new family of linear codes in the Hamming space. Bounds on the minimum distance and the dimension of such codes are given.