In this paper we will study the action of F q n 2 × 2 on the graph of an F q -linear function of F q n to 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 give some examples where this is not the case. We will also connect such a stabilizer to the right idealizer of the rank-metric code defined by the linear function, and give some structural results in the case where the polynomials are partially scattered.
On the stabilizer of the graph of linear functions over finite fields
Smaldore, Valentino;Zanella, Corrado;
2025
Abstract
In this paper we will study the action of F q n 2 × 2 on the graph of an F q -linear function of F q n to 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 give some examples where this is not the case. We will also connect such a stabilizer to the right idealizer of the rank-metric code defined by the linear function, and give some structural results in the case where the polynomials are partially scattered.
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