Affine-invariant strictly cyclic Steiner quadruple systems
Affine-invariant strictly cyclic Steiner quadruple systems
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DOI：
10.1007/s10623-016-0201-z
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发表时间：
2016-04
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影响因子：
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通讯作者：
Xiaonan Lu;Masakazu Jimbo
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文献类型：
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作者：
Xiaonan Lu;Masakazu Jimbo
确定承认特定自同构群的斯坦纳四重系统（SQS）的谱是设计理论研究的一个重要课题。我们考虑在仿射群下不变的严格循环SQS，称为AsSQS。对于实验设计、组测试设计、归档方案设计、认证码设计、CDMA通信光正交码设计等应用，包含循环群的较大自同构群可以有效地完成设计中块的生成和搜索过程，且存储空间和时间更少。本文研究了一类质量问题的构造和存在的必要条件。对于素数，直接构造a建立了一个AsSQS(2p)，假设存在一个图的1因子，其中图由射影特殊线性群PSL（2,p）的生成系统定义。直接构造B给出了一个2色的AsSQS(2p)，假设存在一个特定超图的彩虹1因子。因此，通过对正整子的AsSQS的两个递归构造，我们证明了AsSQS存在，如果AsSQS（2p）的准则满足。我们验证了这一说法，发现assqsexsts适用于所有带正合子的素数和任何正合子。
To determine the spectrum of Steiner quadruple systems (SQS) admitting a specific automorphism group is of great interest in design theory. We consider a strictly cyclic SQS which is invariant under the affine group, called an AsSQS. For the applications of designs of experiments, group testing, filing schemes, authentication codes, and optical orthogonal codes for CDMA communication, etc., a larger automorphism group containing the cyclic group may work efficiently for the procedures of generating and searching blocks in a design with less storage and time. In this paper, constructions and a necessary condition for the existence of an AsSQS are investigated. For a prime, Direct Construction A establishes an AsSQS(2p), provided that a 1-factor of a graph exists, where the graph is defined by using a system of generators of the projective special linear group PSL(2,p). Direct Construction B gives an AsSQS(2p) which is 2-chromatic, provided that a rainbow 1-factor of a specific hypergraph exists. Accordingly, by proposing two recursive constructions of an AsSQSsfor a positive integerm, we prove that an AsSQSexists, if the criteria developed for an AsSQS(2p) are satisfied. We verified the claim and found that an AsSQSexists for every primewithand any positive integerm.