On the 2-ranks of a class of unitals

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• 作者：Rocco Trombetti^a ; ^{rtrombet@unina.it" class="auth_mail" title="E-mail the corresponding author} ; Yue Zhou^a ; ^b ; ^{yue.zhou.ovgu@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：51A45 ; 12K10 ; 51A35 ; 11L05
• 刊名：Finite Fields and Their Applications
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：40
• 期：Complete
• 页码：90-109
• 全文大小：447 K

文摘

Let U_θ${\mathcal{U}}_{\theta }$ be a unital defined in a shift plane of odd order i2" class="mathmlsrc">i2.gif&_user=111111111&_pii=S107157971600040X&_rdoc=1&_issn=10715797&md5=75fc1c3a41375f66ccdeaad19fed5377" title="Click to view the MathML source">q², which are constructed recently in [40]. In particular, when the shift plane is desarguesian, U_θ${\mathcal{U}}_{\theta }$ is a special Buekenhout–Metz unital formed by a union of ovals. We investigate the dimensions of the binary codes derived from U_θ${\mathcal{U}}_{\theta }$. By using Kloosterman sums, we obtain a new lower bound on the aforementioned dimensions which improves Leung and Xiang's result  and . In particular, for q=3^m$q=3^m$, this new lower bound equals $\frac{2}{3}(q^3+q^2-2q)-1$ for even m   and $\frac{2}{3}(q^3+q^2+q)-1$ for odd m.