Automorphism groups of maximum scattered linear sets in finite projective lines

Rank-metric codes have become a standard tool in recent time in the transmission of information theory based upon random networks. This is one of the main reasons why they are widely studied, and why several new constructions recently appeared in literature. Maximum scattered linear sets of the projective line over a finite field give rise to special type of rank-metric codes; i.e. linear maximum rank distance codes (MRD codes, for short) with high value for the minimum distance. Moreover, it is known that if relevant linear MRD codes are equivalent, then the ensuing linear sets are equivalent as well [J. Sheekey. A new family of linear maximum rank distance codes. Adv. Math. Commun. 10(3), 2016]. For this reason, knowledge of the full automorphism group of a maximum linear set of the projective line may play a central role in understanding whether or not a new construction of codes with above mentioned parameters effectively contains, up to equivalence, new examples. The aim of this article is to determine, for any given maximum scattered linear set LU of PG(1,qn) defined by an Fq-subspace U, all possibilities for the structure of its automorphism group. In particular, if the code associated with U has a non-trivial right idealizer which is not too small, then Aut(LU)∩PGL(2,qn) must be cyclic or dihedral.