Partially scattered linearized polynomials and rank metric codes

A linearized polynomial f(x)∈Fqjavax.xml.bind.JAXBElement@26b9a41a[x] is called scattered if for any y,z∈Fqjavax.xml.bind.JAXBElement@43507877, 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.