New maximum scattered linear sets of the projective line

TL;DR

This paper introduces new maximum scattered linear sets of the projective line over finite fields, expanding known examples and linking them to novel MRD-codes with specific algebraic properties.

Contribution

It demonstrates the novelty of certain linear sets for specific parameters and constructs new MRD-codes from these sets, addressing open problems in the field.

Findings

New examples of maximum scattered linear sets for n=6,8

Construction of MRD-codes with dimension 12 and minimum distance 5

Identification of linear sets arising from trinomial polynomials

Abstract

In [2] and [19] are presented the first two families of maximum scattered $\mathbb{F}_q$-linear sets of the projective line $\mathrm{PG}(1,q^n)$. More recently in [23] and in [5], new examples of maximum scattered $\mathbb{F}_q$-subspaces of $V(2,q^n)$ have been constructed, but the equivalence problem of the corresponding linear sets is left open. Here we show that the $\mathbb{F}_q$-linear sets presented in [23] and in [5], for $n=6,8$, are new. Also, for $q$ odd, $q\equiv \pm 1,\,0 \pmod 5$, we present new examples of maximum scattered $\mathbb{F}_q$-linear sets in $\mathrm{PG}(1,q^6)$, arising from trinomial polynomials, which define new $\mathbb{F}_q$-linear MRD-codes of $\mathbb{F}_q^{6\times 6}$ with dimension $12$, minimum distance 5 and middle nucleus (or left idealiser) isomorphic to $\mathbb{F}_{q^6}$.