A new family of maximum scattered linear sets in $\mathrm{PG}(1,q^6)$

TL;DR

This paper introduces a new family of maximum scattered linear sets in projective space, proving their maximality for odd q and deriving new MRD-codes, thus solving an open problem and expanding coding theory.

Contribution

It generalizes previous examples of linear sets to a broader family, establishing their maximum scattered property and producing new MRD-codes, addressing an open research question.

Findings

The new linear sets are maximum scattered when q is odd.

They are mostly new, except for a special case.

The family yields new MRD-codes with parameters (6,6,q;5).

Abstract

We generalize the example of linear set presented by the last two authors in "Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$" (2019) to a more general family, proving that such linear sets are maximum scattered when $q$ is odd and, apart from a special case, they are are new. This solves an open problem posed in "Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$" (2019). As a consequence of Sheekey's results in "A new family of linear maximum rank distance codes" (2016), this family yields to new MRD-codes with parameters $(6,6,q;5)$.