In this paper we will study the action of $\mathbb{ F}_{ q^n}^{ 2 \times 2}$ on the graph of an $\mathbb{ F}_q$-linear function of $\mathbb{ F}_{ q^n}$ into itself. In particular we will see that, under certain combinatorial assumptions, its stabilizer (together with the sum and product of matrices) is a field. We will also see some examples for which this does not happen. Moreover, we will establish a connection between such a stabilizer and the right idealizer of the rank-metric code defined by the linear function and give some structural results in the case in which the polynomials are partially scattered.
Categories: math.CO cs.IT math.IT