论文标题
嵌入式的反式飞机和点型平面中的双重代码和线条的最小重量$ p^2 $
Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order $p^2$
论文作者
论文摘要
射影平面中点与线的发病率矩阵产生的代码的最小重量已有50多年了。令人惊讶的是，即使在Desarguesian案件中，发现非占据订单的射击平面的双重代码的最小重量仍然是一个空旷的问题。 在本文中，我们专注于订单$ p^2 $的投影平面，其中$ p $是主要的，我们将双重代码中的小重量代码单词的存在与嵌入式子类别的存在和{\ em em antipodal Planes}联系起来。在Desarguesian情况下，我们可以通过显示更普遍的结果来排除此类代码词，即没有至少3个对抗阶阶平面可以嵌入到Desarguesian投射平面中。 此外，我们使用组合论证来排除订单$ p^2 $，$ p $ prime的任意投影平面的双重代码中的代码单词的存在，最多使用$ 2p^2-2p+4 $使用两个符号。特别是，这导致了以下结果：Desarguesian投影平面的双代码$ \ MATHRM {PG}（2，P^2）$，$ P \ geq 5 $，至少$ 2P^2-2p+5 $。
The minimum weight of the code generated by the incidence matrix of points versus lines in a projective plane has been known for over 50 years. Surprisingly, finding the minimum weight of the dual code of projective planes of non-prime order is still an open problem, even in the Desarguesian case. In this paper, we focus on the case of projective planes of order $p^2$, where $p$ is prime, and we link the existence of small weight code words in the dual code to the existence of embedded subplanes and {\em antipodal planes}. In the Desarguesian case, we can exclude such code words by showing a more general result that no antipodal plane of order at least 3 can be embedded in a Desarguesian projective plane. Furthermore, we use combinatorial arguments to rule out the existence of code words in the dual code of points and lines of an arbitrary projective plane of order $p^2$, $p$ prime, of weight at most $2p^2-2p+4$ using more than two symbols. In particular, this leads to the result that the dual code of the Desarguesian projective plane $\mathrm{PG}(2,p^2)$, $p\geq 5$, has minimum weight at least $2p^2-2p+5$.
