A new family of MRD-codes

TL;DR

This paper introduces a new family of maximum rank distance (MRD) codes derived from maximum scattered linear sets in projective spaces, expanding the known classes of such codes with specific parameters for certain dimensions and field sizes.

Contribution

It presents a novel construction of MRD-codes from linear sets of pseudoregulus type, providing new examples with specific parameters for dimensions 6 and 8.

Findings

New MRD-codes with parameters (6,6,q;5) for q>2.

New MRD-codes with parameters (8,8,q;7) for q odd.

Identification of maximum scattered linear sets of pseudoregulus type.

Abstract

We introduce a family of linear sets of $\mathrm{PG}(1,q^{2n})$ arising from maximum scattered linear sets of pseudoregulus type of $\mathrm{PG}(3,q^{n})$. For $n=3,4$ and for certain values of the parameters we show that these linear sets of $\mathrm{PG}(1,q^{2n})$ are maximum scattered and they yield new MRD-codes with parameters $(6,6,q;5)$ for $q>2$ and with parameters $(8,8,q;7)$ for $q$ odd.