Partially scattered linearized polynomials and rank metric codes

Abstract

A linearized polynomial f (x) ∈ F q n [x] is called scattered if for any y, z ∈ F q n , the condition zf (y) − yf (z) = 0 implies that y and z are F q -linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are defined and investigated. Let t be a nontrivial positive divisor of n. By weakening the property defining a scattered linearized polynomial, L-q t -partially scattered and R-q t -partially scattered linearized polynomials are introduced in such a way that the scattered linearized polynomials are precisely those which are both L-q t -and R-q t -partially scattered. They determine linear sets and maximum rank distance codes whose properties are described in this paper.

Exceptional scattered polynomials

“…In [24], the authors weaken the property of being scattered for a polynomial as in Equations ( 2) and (3). In the following we resume the results contained in [24] which will be useful for our purposes.…”

“…In [24], the authors weaken the property of being scattered for a polynomial as in Equations ( 2) and (3). In the following we resume the results contained in [24] which will be useful for our purposes. For any F q -linearized polynomial f (x) ∈ F q n [x] and any ρ ∈ F * q n , define f ρ (x) := f (ρx) − ρf (x).…”

“…Remark 5.7. The family presented in Proposition 5.1 contains the examples of R-q tpartially scattered given in [24]:…”

“…However, their classification is still unknown when the index is greater than 1. In this paper we investigate the exceptional scatteredness of a polynomial by considering separately the exceptionality of two weaker properties defined in [24], namely the L-q t -partial scatteredness and the R-q t -partial scatteredness.…”

“…Some results on, and characterizations of L-and R-q t -partially scattered polynomials have been provided in [24]; see Section 2. In this paper, we start the investigation of such properties by those families that contain the known examples of exceptional scattered polynomials, namely the monomials x q u of index 0 and the LP polynomials x + δx q 2s of index s. In this way we obtain exceptional L-and R-q t -partially scattered monomials which are not scattered, and other results for Land R-q t -partially scattered polynomials of LP type; see Section 3.…”