Rank-metric codes, linear sets, and their duality

Abstract

In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of Sheekey (Adv Math Commun 10:475–488, 2016, Sect. 5) on linear sets on a projective line. We extend this to linear sets in arbitrary dimension, giving the connection between two constructions for linear sets defined in Lunardon (J Comb Theory Ser A 149:1–20, 2017). Finally, we then exploit this connection by using the MacWilliams identities to obtain information about the possible weight distribution of a linear set of rank n on a projective line $$\mathrm {PG}(1,q^n)$$.