2023/07/24 by Daniele Bartoli, Giovanni Longobardi, Bartoli, Daniele +5
Computer Science · Mathematics · #05B25 #06E30 #11T06 #51E20 #51E22 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2307.12829
openalex publication_date 2023/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In [B. Csajbók, G. Marino and F. Zullo: New maximum scattered linear sets of the projective line, Finite Fields Appl. 54 (2018), 133-150; G. Marino, M. Montanucci and F. Zullo: MRD-codes arising from the trinomial xq+xq3+cxq5∈\mathbbFq6[x], Linear Algebra Appl. 591 (2020), 99-114], the authors proved that the trinomial fc(X)=Xq+X^q3+cX^q5 of \mathbbFq6[X] is scattered under the assumptions that q is odd and c2+c=1. They also explicitly observed that this is false when q is even. In this paper, we provide a different set of conditions on c for which this trinomial is scattered in the case of even q. Using tools of algebraic geometry in positive characteristic, we show that when q is even and sufficiently large, there are roughly q3 elements c ∈ \mathbbFq6 such that fc(X) is scattered. Also, we prove that the corresponding MRD-codes and \mathbbFq-linear sets of PG(1,q6) are not equivalent to the previously known ones.