Linearized trinomials with maximum kernel

12/29/2020
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by   Paolo Santonastaso, et al.
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Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let q be a prime power, n be a positive integer and Ļƒ be a generator of Gal(š”½_q^nš”½_q). In this paper we provide closed formulas for the coefficients of a Ļƒ-trinomial f over š”½_q^n which ensure that the dimension of the kernel of f equals its Ļƒ-degree, that is linearized polynomials with maximum kernel. As a consequence, we present explicit examples of linearized trinomials with maximum kernel and characterize those having Ļƒ-degree 3 and 4. Our techniques rely on the tools developed in [22]. Finally, we apply these results to investigate a class of rank metric codes introduced in [8], to construct quasi subfield polynomials and cyclic subspace codes, obtaining new solutions to the conjecture posed in [35].

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