Affine-invariant quadruple systems-猫眼课题宝

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2014

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Xiaonan Lu

设t，v，k，λ为满足v > k > t的正整数。一个t-（v，k，λ）设计是一个有序对（V，B），其中V是v个点的有限集，B是V的k个子集的集合，比如块，使得V的每个t-子集恰好出现在B中的λ个块中。下面我们简单地写t-设计。一个3-（v，4，1）设计称为Steiner四重系统，记为SQS（v）。已知SQS（v）存在当且仅当v ≠ 2，4（mod 6）（见[9]）。当λ > 1时，一个3-（v，4，λ）设计称为λ-重四重系统，简称为λ-重QS（v）。t-设计（V，B）的自同构群G是定义在V上的置换群，使得B不变。对固定块B ∈ B，B在G下的轨道为OG（B）= {B| g ∈ G}。因此，B可以被划分为G下的轨道，即G-轨道。此外，如果轨道O的基数等于G的阶，则O被称为满的，否则，短的。O中的任何块都可以看作是轨道的基块。特别地，一个t-（v，k，λ）-设计称为循环的，如果它允许一个v阶循环群Cv作为它的自同构。一个CV-轨道称为循环轨道。在不失一般性的情况下，我们确定的点集的循环t-设计的加法组Zv = Z/vZ，整数模v。此外，循环t-设计被称为严格循环的，如果所有的循环轨道是满的。在下文中，我们用CSQS表示循环SQS，用sSQS表示严格循环SQS。CSQS（v）和sSQS（v）存在的必要条件分别是v = 2，4（mod 6）和v = 2，10（mod 24）（参见[12]）。Köhler在sSQS上的工作[12]建立了sSQS和以他的名字命名的“Köhler图”的11个因子之间的联系。西蒙[23] [24]对科勒工作的一些方法检查了“科勒图”的1-因子对于相当多的容许参数的存在性。Piotrowski [22]构造了sSQS（2 p），允许二面角

Let t, v, k, λ be positive integers satisfying v > k > t. A t-(v, k, λ) design is an ordered pair (V,B), where V is a finite set of v points, B is a collection of k-subsets of V , say blocks, such that every t-subset of V occurs in exactly λ blocks in B. In what follows we simply write t-designs. A 3-(v, 4, 1) design is called a Steiner quadruple system and denoted by SQS(v). It is known that an SQS(v) exists if and only if v ≡ 2, 4 (mod 6) (see [9]). For λ > 1, a 3-(v, 4, λ) design is called a λ-fold quadruple system and denoted by λ-fold QS(v) for short. An automorphism group G of a t-design (V,B) is a permutation group defined on V which leaves B invariant. For a fixed block B ∈ B, the orbit of B under G is OG(B) = {B | g ∈ G}. Thus, B can be partitioned into orbits under G, say G-orbits. Moreover, if the cardinality of an orbit O equals to the order of G, then O is said to be full, otherwise, short. Any block in O can be regarded as a base block of the orbit. In particular, a t-(v, k, λ)-design is said to be cyclic if it admits a cyclic group Cv of order v as its automorphism. A Cv-orbit is called a cyclic orbit. Without loss of generality, we identify the point set of a cyclic t-design with the additive group of Zv = Z/vZ, the integers modulo v. Furthermore, a cyclic t-design is said to be strictly cyclic, if all cyclic orbits are full. In what follows, we denote a cyclic SQS by CSQS, a strictly cyclic SQS by sSQS. The necessary conditions for the existence of a CSQS(v) and an sSQS(v) are v ≡ 2, 4 (mod 6) and v ≡ 2, 10 (mod 24) respectively (see [12]). The work on sSQS by Köhler [12] established a connection between sSQS and 1factors of “Köhler graphs” named after him. Some approaches to Köhler’s work by Siemon [23] [24] checked the existence of 1-factors of “Köhler graphs” for quite a few admissible parameters. Piotrowski [22] constructed sSQS(2p) admitting the dihedral

Affine-invariant quadruple systems

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