New maximum scattered linear sets of the projective line

Abstract

In [2] and [19] are presented the first two families of maximum scattered F q -linear sets of the projective line PG(1, q n ). More recently in [23] and in [5], new examples of maximum scattered F q -subspaces of V (2, q n ) have been constructed, but the equivalence problem of the corresponding linear sets is left open.Here we show that the F q -linear sets presented in [23] and in [5], for n = 6, 8, are new. Also, for q odd, q ≡ ±1, 0 (mod 5), we present new examples of maximum scattered F q -linear sets in PG(1, q 6 ), arising from trinomial polynomials, which define new F q -linear MRD-codes of F 6×6 q with dimension 12, minimum distance 5 and middle nucleus (or left idealiser) isomorphic to F q 6 .

Vertex properties of maximum scattered linear sets of PG(1,qn)

“…We show that for q ≤ 17 and q ≡ 0 (mod 5) it is also new. For q ≡ 0 (mod 5) we will prove in Proposition 5.5 that L is equivalent to the linear set defined in [9].…”

“…This is a polynomial identity in x σ and hence we have the Equations (9). Now, suppose that there exist σ ∈ Aut(F q 6 ) and an invertible matrix a b c d such that for each x ∈ F q 6 there exists z ∈ F q 6 satisfying a b c d…”

A characterization of the number of roots of linearized and projective polynomials in the field of coefficients

“…Further constructions have been obtained in the case n = 6, 8 [3], [5], again using properties of σ-linearized polynomials.…”

A condition for scattered linearized polynomials involving Dickson matrices

“…In particular cases, the condition N q n /q (δ) = 1 has been proved to be necessary for f (x) to be scattered [2,10,11,22]. In section 3 it will proved that actually it is necessary for any n and s. Further examples of scattered q-polynomials are given in [6,5,14,22]. All of them are with respect to n = 6 or n = 8.…”