论文标题
超规范的Steiner 2设计
Super-regular Steiner 2-designs
论文作者
论文摘要
在Abelian Group $ G $（简短，$ G $ - addive）下，设计是添加的，如果达到同构，其点集包含在$ G $中，并且每个块的元素总和为零。对于某些$ g $的$ g $ addive，唯一已知的Steiner 2设计具有块大小，它是Prime Power或Prime Power加一个块。实际上，它们是仿射空间$ ag（n，q）$的点线设计，投影平面的点线设计$ pg（2，q）$以及投影型空间$ pg（n，2）$的点线设计。为了找到新的示例，可能既不是主要功率，也不是Prime Power Plus One的块大小，我们寻找严格的$ G $ - addive（点集正好是$ G $）和$ g $ regormular（任何块的翻译也是任何块，也是一个块））。这些设计将称为\ break“ $ g $ -super-regular”。我们的主要结果是，在$ v $中存在无限的$ V $值，而每当$ k $既不是$ 2^n3 \ geq12 $的$ k $，nove $ 2 $ - $（v，v，k，1）$设计。 $ k \ equiv2 $（mod 4）是一个确定的异常，而$ k = 2^n3 \ geq12 $目前是可能的例外。我们还发现超规范$ 2 $ - $（p^n，p，1）$ $ p \ in \ in \ {5,7 \} $和$ n \ geq3 $与$ ag（n，p）$的点线设计不是同构的$ n \ geq3 $。
A design is additive under an abelian group $G$ (briefly, $G$-additive) if, up to isomorphism, its point set is contained in $G$ and the elements of each block sum up to zero. The only known Steiner 2-designs that are $G$-additive for some $G$ have block size which is either a prime power or a prime power plus one. Indeed they are the point-line designs of the affine spaces $AG(n,q)$, the point-line designs of the projective planes $PG(2,q)$, and the point-line designs of the projective spaces $PG(n,2)$. In the attempt to find new examples, possibly with a block size which is neither a prime power nor a prime power plus one, we look for Steiner 2-designs which are strictly $G$-additive (the point set is exactly $G$) and $G$-regular (any translate of any block is a block as well) at the same time. These designs will be called\break "$G$-super-regular". Our main result is that there are infinitely many values of $v$ for which there exists a super-regular, and therefore additive, $2$-$(v,k,1)$ design whenever $k$ is neither singly even nor of the form $2^n3\geq12$. The case $k\equiv2$ (mod 4) is a definite exception whereas $k=2^n3\geq12$ is at the moment a possible exception. We also find super-regular $2$-$(p^n,p,1)$ designs with $p\in\{5,7\}$ and $n\geq3$ which are not isomorphic to the point-line design of $AG(n,p)$.
