A Large Family of Maximum Scattered Linear Sets of $${{\,\mathrm{{PG}}\,}}(1,q^n)$$ and Their Associated MRD Codes

The concept of linear set in projective spaces over finite fields was introduced by Lunardon 1999 and it plays central roles the study blocking sets, semifields, rank-distance codes etc. A with largest possible cardinality rank is called maximum scattered. Despite two decades study, there are only a limited number scattered sets line $\mathrm{PG}(1,q^n)$. In this paper, we provide large family new $\mathrm{PG}(1,q^n)$ for any even $n\geq 6$ odd $q$. particular, relevant contains at least \[ \begin{cases} \left\lfloor\frac{q^t+1}{8rt}\right\rfloor,& \text{ if }t\not\equiv 2\pmod{4};\\[8pt] \left\lfloor\frac{q^t+1}{4rt(q^2+1)}\right\rfloor,& }t\equiv 2\pmod{4}, \end{cases} \] inequivalent members given $q=p^r$ $n=2t>8$, where $p=\mathrm{char}(\mathbb{F}_q)$. This great improvement previous results: $q$ $n>8$, all classes known so far, smaller than $q^2$. Moreover, show that arising from constructed sets.