论文标题
与Desarguesian卵形相关的线性代码，$ q^+（7，q）$
Linear codes associated with the Desarguesian ovoids in $Q^+(7,q)$
论文作者
论文摘要
正交极空间中的Desarguesian卵形$ q^+（7，q）$ d $ q $甚至首先是坎托（Kantor）通过检查$ 8 $二维绝对不可减少的模块化表示的$ \ text {pgl}（2，q^3）$。我们为$ Q $的所有主要功率值调查了此模块。最短的$ \ text {pgl}（2，q^3）$ - 轨道$ o $为$ q^+（7，q）的desarguesian ovoid for $ q $，众所周知，它可以为奇数〜$ q $提供完全的偏卵形$ w（7，q）。我们确定$ o $的超平面部分。作为推论，我们获得了参数$ [q^3+1,8，q^3-q^2-q] _q $以及关联的$ \ mathbb {f} _q $ - 线性代码$ c_o $和参数$和参数$ [q^3+1，q^3-7,7,7,5] _Q $ q_ $ c_o $ c_o Q^$ c_o q of的权重分布。我们还表明，这两个代码$ C_O $和$ C_O^\ PERP $对于所有$ Q $的所有质量功率值都是长度最佳的。
The Desarguesian ovoids in the orthogonal polar space $Q^+(7,q)$ with $q$ even have first been introduced by Kantor by examining the $8$-dimensional absolutely irreducible modular representations of $\text{PGL}(2,q^3)$. We investigate this module for all prime power values of $q$. The shortest $\text{PGL}(2,q^3)$-orbit $O$ gives the Desarguesian ovoid in $Q^+(7,q)$ for even $q$ and it is known to give a complete partial ovoid of the symplectic polar space $W(7,q)$ for odd~$q$. We determine the hyperplane sections of $O$. As a corollary, we obtain the parameters $[q^3+1,8,q^3-q^2-q]_q$ and the weight distribution of the associated $\mathbb{F}_q$-linear code $C_O$ and the parameters $[q^3+1,q^3-7,5]_q$ of the dual code $C_O^\perp$ for $q \ge 4$. We also show that both codes $C_O$ and $C_O^\perp$ are length-optimal for all prime power values of $q$.
