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Maximum scattered subspaces are not only objects of intrinsic interest infinite 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 scatteredsubspaces in $\mathbb{F}_{q^6} \times \mathbb{F}_{q^6}$ arising frompolynomials 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 authorsconjectured that the portion of new and inequivalent maximum scatteredsubspaces obtained in this way is quite large. In this paper first we findnecessary and sufficient conditions for $b$ to obtain a maximum scatteredsubspace. Such conditions were found independently with different techniquesalso by Polverino and Zullo 2019. Then we prove the conjecture on the number ofnew and inequivalent maximum scattered subspaces of this family.

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