On the additivity of block designs

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• 作者：Andrea Caggegi ; Giovanni Falcone ; Marco Pavone
• 关键词：Block designs ; Symmetric block designs ; Hadamard designs ; Steiner triple systems
• 刊名：Journal of Algebraic Combinatorics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：45
• 期：1
• 页码：271-294
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Convex and Discrete Geometry; Order, Lattices, Ordered Algebraic Structures; Computer Science, general; Group Theory and Generalizations;
• 出版者：Springer US
• ISSN：1572-9192
• 卷排序：45

文摘

We show that symmetric block designs \({\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})\) can be embedded in a suitable commutative group \({\mathfrak {G}}_{\mathcal {D}}\) in such a way that the sum of the elements in each block is zero, whereas the only Steiner triple systems with this property are the point-line designs of \({\mathrm {PG}}(d,2)\) and \({\mathrm {AG}}(d,3)\). In both cases, the blocks can be characterized as the only k-subsets of \(\mathcal {P}\) whose elements sum to zero. It follows that the group of automorphisms of any such design \(\mathcal {D}\) is the group of automorphisms of \({\mathfrak {G}}_\mathcal {D}\) that leave \(\mathcal {P}\) invariant. In some special cases, the group \({\mathfrak {G}}_\mathcal {D}\) can be determined uniquely by the parameters of \(\mathcal {D}\). For instance, if \(\mathcal {D}\) is a 2-\((v,k,\lambda )\) symmetric design of prime order p not dividing k, then \({\mathfrak {G}}_\mathcal {D}\) is (essentially) isomorphic to \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\), and the embedding of the design in the group can be described explicitly. Moreover, in this case, the blocks of \(\mathcal {B}\) can be characterized also as the v intersections of \(\mathcal {P}\) with v suitable hyperplanes of \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\).