Linearized polynomials over finite fields revisited

Abstract

We give new characterizations of the algebra L n (F q n ) formed by all linearized polynomials over the finite field F q n after briefly surveying some known ones. One isomorphism we construct is between L n (F q n ) and the composition algebra F ∨ q n ⊗ Fq F q n . The other isomorphism we construct is between L n (F q n ) and the so-called Dickson matrix algebra D n (F q n ). We also further study the relations between a linearized polynomial and its associated Dickson matrix, generalizing a well-known criterion of Dickson on linearized permutation polynomials. Adjugate polynomial of a linearized polynomial is then introduced, and connections between them are discussed. Both of the new characterizations can bring us more simple approaches to establish a special form of representations of linearized polynomials proposed recently by several authors. Structure of the subalgebra L n (F q m ) which are formed by all linearized polynomials over a subfield F q m of F q n where m|n are also described.

New semifields and new MRD codes from skew polynomial rings

“…Clearly any linearized polynomial defines a ‐linear map of , . If we define the product of linearized polynomials to be composition, and identify with , then we get a ring isomorphism (see, for example, ): Note that from this correspondence, we may say that a code is ‐linear if the corresponding set of linearized polynomials is closed under multiplication by . We can also move easily from linearized polynomials to vectors in , simply by evaluating a polynomial on a set of elements of , linearly independent over .…”

Several Classes of Quadratic Ternary Bent, Near-Bent and 2-Plateaued Functions

“…Recently, the compositional inverse of of some linearized permutation polynomials have been discovered. For detailed information, the reader is referred to [11,[30][31][32].…”

New classes of quadratic bent functions in polynomial forms

“…Very recently, more relations between properties of L(x) and D L (called the associated Dickson matrix of L(x)) are found by the author and Liu [11].…”