NDLI: On the additivity of block designs

Content Provider	Springer Nature Link
Author	Caggegi, Andrea Falcone, Giovanni Pavone, Marco
Copyright Year	2016
Abstract	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 $$({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}$$ .
Ending Page	294
Page Count	24
Starting Page	271
File Format	PDF
ISSN	09259899
e-ISSN	15729192
Journal	Journal of Algebraic Combinatorics
Issue Number	1
Volume Number	45
Language	English
Publisher	Springer US
Publisher Date	2016-09-02
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Hadamard designs Symmetric block designs Convex and Discrete Geometry Computer Science Finite geometries Block designs Order, Lattices, Ordered Algebraic Structures Group Theory and Generalizations Matrices (incidence, Hadamard, etc.) Combinatorics Steiner triple systems Triple systems
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Algebra and Number Theory

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