t1/2 - Designs
t1/2 - Designs
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DOI：
10.1016/0097-3165(80)90067-9
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发表时间：
1980
期刊：
影响因子：
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通讯作者：
A. Neumaier
中科院分区：
文献类型：
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作者：
A. Neumaier
每个(t+1)设计B满足(+)如果T是t点的集合，B是B的块，则标志(x，B)的个数α(T，B)具有x∉T，xϵB，T∪{x}⊆A仅取决于|T∩B|。具有性质(+)的t设计称为t12设计。最有趣的一般t-设计类是t12-设计：Hadamard 3-设计是312-设计，对称2-设计是212-设计，对偶2-设计、横截设计和部分几何是1122-设计；事实上，1122-设计分享了部分几何的大部分性质。详细研究了1 1 2-设计，并研究了它们与强正则图的关系。结果表明，t1~2-设计在求导、求余、求补方面与t-设计相似。利用t12-设计给出了部分几何、广义四边形、对称2-设计和Hadamard 3-设计的各种刻画。最后证明了t-⩾为4的t12-设计已经是(t+1)-设计。
Abstract Every (t+ 1)-design B satisfies (+) If T is a set of t points, and B a block of B then the number α (T, B) of flags (x, A) with x∉ T, x ϵ B, T∪{x}⊆ A depends only on| T∩ B|. A t-design with property (+) is called a t 1 2-design. The most interesting general classes of t-designs are t 1 2-designs: Hadamard 3-designs are 3 1 2-designs, symmetric 2-designs are 2 1 2-designs, and dual 2-designs, transversal designs, and partial geometries are 1 1 2-designs; in fact, 1 1 2-designs share most properties of partial geometries. 1 1 2-designs are studied in detail, and their connection with strongly regular graphs is investigated. It is shown that t 1 2-designs behave like t-designs with respect to derivation, residuals, and complementation. Various characterizations of partial geometries, generalized quadrangles, symmetric 2-designs, and Hadamard 3-designs are given in terms of t 1 2-designs. The paper ends with a proof that t 1 2-designs with t⩾ 4 are already (t+ 1)-designs.