On a conjecture about maximum scattered subspaces of $\mathbb{F}_{q^6}\times \mathbb{F}_{q^6}$

TL;DR

This paper investigates a family of maximum scattered subspaces in finite geometry, providing necessary and sufficient conditions for their construction and confirming a conjecture about their abundance and inequivalence.

Contribution

It establishes precise criteria for constructing these subspaces and proves the conjecture that many new inequivalent examples exist within this family.

Findings

Necessary and sufficient conditions for parameters to produce maximum scattered subspaces.

Confirmation of the conjecture on the large number of inequivalent subspaces.

Independent derivations of conditions by different researchers.

Abstract

Maximum scattered subspaces are not only objects of intrinsic interest in finite geometry but also powerful tools for the construction of MRD-codes, projective two-weight codes, and strongly regular graphs. In 2018 Csajb\'ok, Marino, Polverino, and Zanella introduced a new family of maximum scattered subspaces in $\mathbb{F}_{q^6} \times \mathbb{F}_{q^6}$ arising from polynomials of type $f_b(x)=bx^q+x^{q^4}$ for certain choices of $b \in \mathbb{F}_{q^6}$. Throughout characterizations for $f_{b_2}(x)$ and $f_{b_1}(x)$ giving rise to equivalent maximum scattered subspaces, the authors conjectured that the portion of new and inequivalent maximum scattered subspaces obtained in this way is quite large. In this paper first we find necessary and sufficient conditions for $b$ to obtain a maximum scattered subspace. Such conditions were found independently with different techniques also by Polverino and Zullo 2019. Then we prove the conjecture on the number of new and inequivalent maximum scattered subspaces of this family.