Partially scattered linearized polynomials and rank metric codes

A linearized polynomial f(x)∈Fqn[x] is called scattered if for any y,z∈Fqn, the condition zf(y)−yf(z)=0 implies that y and z are Fq-linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are provided and investigated. Let t be a nontrivial positive d...

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Bibliographic Details
Published inFinite fields and their applications Vol. 76; p. 101914
Main Authors Longobardi, Giovanni, Zanella, Corrado
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.12.2021
Subjects
Finite field
Finite projective space
Linear set
Linearized polynomial
MRD-code
Rank metric code
Subgeometry
MRD-code
Finite field
Finite projective space
Subgeometry
51E20
Linear set
Rank metric code
51E22
05B25
Linearized polynomial
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ISSN1071-5797
1090-2465
DOI10.1016/j.ffa.2021.101914

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Summary:A linearized polynomial f(x)∈Fqn[x] is called scattered if for any y,z∈Fqn, the condition zf(y)−yf(z)=0 implies that y and z are Fq-linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are provided and investigated. Let t be a nontrivial positive divisor of n. By weakening the property defining a scattered linearized polynomial, L-qt-partially scattered and R-qt-partially scattered linearized polynomials are introduced in such a way that the scattered linearized polynomials are precisely those which are both L-qt- and R-qt-partially scattered. Also, connections between partially scattered polynomials, linear sets and rank metric codes are exhibited.
ISSN:1071-5797
1090-2465
DOI:10.1016/j.ffa.2021.101914