论文标题
关于$ \ mathbb {f} _ {q^6} \ times \ mathbb {f} _ {q^6} $的猜测最大分散子空间
On a conjecture about maximum scattered subspaces of $\mathbb{F}_{q^6}\times \mathbb{F}_{q^6}$
论文作者
论文摘要
最大散射子空间不仅是有限几何形状中固有兴趣的对象，而且是用于构建MRD代码，投影两次重量代码和强烈规则图的强大工具。在2018年，CSAJBók，Marino，Polverino和Zanella在$ \ Mathbb {f} _ {q^6} \ times \ times \ Mathbb {f} _ {q^6} $中引起了一个新的最大分散子空间家族，以$ f_b（x）$ f_b（x） \ in \ Mathbb {f} _ {q^6} $。在整个特征中，$ f_ {b_2}（x）$和$ f_ {b_1}（x）$产生了等效的最大散射子空间，作者认为，以这种方式获得的新的和不等的最大散射子空间的一部分很大。首先，在本文中，我们发现$ b $获得最大分散子空间的必要条件。 Polverino和Zullo 2019也通过不同的技术独立发现了这种条件。然后，我们证明了该家族的新最大散射子空间的猜想。
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 $\mathbb{F}_{q^6} \times \mathbb{F}_{q^6}$ arising from polynomials of type $f_b(x)=bx^q+x^{q^4}$ for certain choices of $b \in \mathbb{F}_{q^6}$. Throughout characterizations for $f_{b_2}(x)$ and $f_{b_1}(x)$ 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.
