A generalization of the normal rational curve in $$\mathop {\mathrm{PG}}(d,q^n)$$PG(d,qn) and its associated non-linear MRD codes

Let A and B be two points of $$\mathop {\mathrm{PG}}(d,q^n)$$PG(d,qn) and let $$\Phi $$Φ be a collineation between the stars of lines with vertices A and B, that does not map the line AB into itself. In this paper we prove that if $$d=2$$d=2 or $$d\ge 3$$d≥3 and the lines $$\Phi ^{-1}(AB), AB, \Phi (AB) $$Φ-1(AB),AB,Φ(AB) are not in a common plane, then the set $$\mathcal{C}$$C of points of intersection of corresponding lines under $$\Phi $$Φ is the union of $$q-1$$q-1 scattered $${\mathbb {F}}_{q}$$Fq-linear sets of rank n together with $$\{A,B\}$${A,B}. As an application we will construct, starting from the set $$\mathcal{C}$$C, infinite families of non-linear $$(d+1, n, q;d-1)$$(d+1,n,q;d-1)-MRD codes, $$d\le n-1$$d≤n-1, generalizing those recently constructed in Cossidente et al. (Des Codes Cryptogr 79:597–609, 2016) and Durante and Siciliano (Electron J Comb, 2017).