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The concept of linear set in projective spaces over finite fields wasintroduced by Lunardon in 1999 and it plays central roles in the study ofblocking sets, semifields, rank-distance codes and etc. A linear set with thelargest possible cardinality and rank is called maximum scattered. Despite twodecades of study, there are only a limited number of maximum scattered linearsets of a line $\mathrm{PG}(1,q^n)$. In this paper, we provide a large familyof new maximum scattered linear sets over $\mathrm{PG}(1,q^n)$ for any even$n\geq 6$ and odd $q$. In particular, the relevant family contains at least  \[ \begin{cases}  \left\lfloor\frac{q^t+1}{8rt}\right\rfloor,& \text{ if }t\not\equiv2\pmod{4};\\[8pt]  \left\lfloor\frac{q^t+1}{4rt(q^2+1)}\right\rfloor,& \text{ if }t\equiv2\pmod{4},  \end{cases} \]  inequivalent members for given $q=p^r$ and $n=2t>8$, where$p=\mathrm{char}(\mathbb{F}_q)$. This is a great improvement of previousresults: for given $q$ and $n>8$, the number of inequivalent maximum scatteredlinear sets of $\mathrm{PG}(1,q^n)$ in all classes known so far, is smallerthan $q^2$. Moreover, we show that there are a large number of new maximumrank-distance codes arising from the constructed linear sets.

A large family of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$ and their associated MRD codes