On a conjecture about maximum scattered subspaces of \mathbbF_q⁶× \mathbbF_q⁶

2020/04/27 by Daniele Bartoli, Bence Csajbók, Bartoli, Daniele +3
Computer Science · Mathematics · #Advanced Algebra and Geometry #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2004.13101

openalex publication_date 2020/04/27 · openalex created_date 2020/05/13 · openalex updated_date 2026/07/28

Abstract

Maximum scattered subspaces are not only objects of intrinsic interest in finite geometry but also powerful tools for the construction of MRD-codes, projective two-weight codes, and strongly regular graphs. In 2018 Csajbók, Marino, Polverino, and Zanella introduced a new family of maximum scattered subspaces in \mathbbF_q⁶ × \mathbbF_q⁶ arising from polynomials of type f_b(x)=bx^q+x^q4 for certain choices of b ∈ \mathbbF_q⁶. Throughout characterizations for f_b2(x) and f_b1(x) giving rise to equivalent maximum scattered subspaces, the authors conjectured that the portion of new and inequivalent maximum scattered subspaces obtained in this way is quite large. In this paper first we find necessary and sufficient conditions for b to obtain a maximum scattered subspace. Such conditions were found independently with different techniques also by Polverino and Zullo 2019. Then we prove the conjecture on the number of new and inequivalent maximum scattered subspaces of this family.

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