### Abstract

Original language | English |
---|---|

Pages (from-to) | 271-294 |

Number of pages | 24 |

Journal | Journal of Algebraic Combinatorics |

Volume | Volume 45, Issue 1 |

Publication status | Published - 2017 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics

### Cite this

*Journal of Algebraic Combinatorics*,

*Volume 45, Issue 1*, 271-294.

**On the additivity of block designs.** /.

Research output: Contribution to journal › Article

*Journal of Algebraic Combinatorics*, vol. Volume 45, Issue 1, pp. 271-294.

}

TY - JOUR

T1 - On the additivity of block designs

AU - Pavone, Marco

AU - Falcone, Giovanni

PY - 2017

Y1 - 2017

N2 - 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}.

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}.

UR - http://hdl.handle.net/10447/201793

M3 - Article

VL - Volume 45, Issue 1

SP - 271

EP - 294

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

ER -