On the additivity of block designs

Andrea Caggegi, Giovanni Falcone, Marco Pavone

Risultato della ricerca: Article

Abstract

We show that symmetric block designs D=(P, B) can be embedded in a suitable commutative group G_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 PG(d,2) and AG(d,3). In both cases, the blocks can be characterized as the only k-subsets of P whose elements sum to zero. It follows that the group of automorphisms of any such design D is the group of automorphisms of G_ D that leave P invariant. In some special cases, the group G_D can be determined uniquely by the parameters of D. For instance, if D is a 2- (v,k,\lambda ) symmetric design of prime order p not dividing k, then G_D is (essentially) isomorphic to (Z/pZ)^{{v-1}/2}, and the embedding of the design in the group can be described explicitly. Moreover, in this case, the blocks of B can be characterized also as the v intersections of P with v suitable hyperplanes of (Z/pZ)^{{v-1}/2}.
Lingua originaleEnglish
pagine (da-a)271-294
Numero di pagine24
RivistaJournal of Algebraic Combinatorics
VolumeVolume 45, Issue 1
Stato di pubblicazionePublished - 2017

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Block Design
Additivity
Symmetric Design
Automorphisms
Steiner Triple System
Zero
Hyperplane
Isomorphic
Intersection
Subset
Invariant
Line
Design

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics

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On the additivity of block designs. / Caggegi, Andrea; Falcone, Giovanni; Pavone, Marco.

In: Journal of Algebraic Combinatorics, Vol. Volume 45, Issue 1, 2017, pag. 271-294.

Risultato della ricerca: Article

Caggegi, A, Falcone, G & Pavone, M 2017, 'On the additivity of block designs', Journal of Algebraic Combinatorics, vol. Volume 45, Issue 1, pagg. 271-294.
Caggegi A, Falcone G, Pavone M. On the additivity of block designs. Journal of Algebraic Combinatorics. 2017;Volume 45, Issue 1:271-294.
Caggegi, Andrea ; Falcone, Giovanni ; Pavone, Marco. / On the additivity of block designs. In: Journal of Algebraic Combinatorics. 2017 ; Vol. Volume 45, Issue 1. pagg. 271-294.
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AB - We show that symmetric block designs D=(P, B) can be embedded in a suitable commutative group G_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 PG(d,2) and AG(d,3). In both cases, the blocks can be characterized as the only k-subsets of P whose elements sum to zero. It follows that the group of automorphisms of any such design D is the group of automorphisms of G_ D that leave P invariant. In some special cases, the group G_D can be determined uniquely by the parameters of D. For instance, if D is a 2- (v,k,\lambda ) symmetric design of prime order p not dividing k, then G_D is (essentially) isomorphic to (Z/pZ)^{{v-1}/2}, and the embedding of the design in the group can be described explicitly. Moreover, in this case, the blocks of B can be characterized also as the v intersections of P with v suitable hyperplanes of (Z/pZ)^{{v-1}/2}.

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