Search results for " Hadamard designs"

showing 3 items of 3 documents

Additivity of affine designs

2020

We show that any affine block design $$\mathcal{D}=(\mathcal{P},\mathcal{B})$$ is a subset of a suitable commutative group $${\mathfrak {G}}_\mathcal{D},$$ with the property that a k-subset of $$\mathcal{P}$$ is a block of $$\mathcal{D}$$ if and only if its k elements sum up to zero. As a consequence, the group of automorphisms of any affine design $$\mathcal{D}$$ is the group of automorphisms of $${\mathfrak {G}}_\mathcal{D}$$ that leave $$\mathcal P$$ invariant. Whenever k is a prime p,  $${\mathfrak {G}}_\mathcal{D}$$ is an elementary abelian p-group.

Algebra and Number Theory010102 general mathematics0102 computer and information sciencesAutomorphism01 natural sciencesCombinatoricsKeywords Affine block designs · Hadamard designs · Additive designs · Mathieu group M11010201 computation theory & mathematicsSettore MAT/05 - Analisi MatematicaAdditive functionDiscrete Mathematics and CombinatoricsAffine transformationSettore MAT/03 - Geometria0101 mathematicsInvariant (mathematics)Abelian groupMathematics
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On the representations in GF(3)^4 of the Hadamard design H_11

2020

In this paper we study the representations of the 2-(11,5,2) Hadamard design H_11 = (P,B) as a set of eleven points in the 4-dimensional vector space GF(3)^4, under the conditions that the five points in each block sum up to zero, and dim ‹P› = 4. We show that, up to linear automorphism, there exist precisely two distinct, linearly nonisomorphic representations, and, in either case, we characterize the family S of all the 5-subsets of P whose elements sum up to zero. In both cases, S properly contains the family of blocks B, thereby showing that a previous result on the representations of H_11 in GF(3)^5 cannot be improved.

Block designs Hadamard designsSettore MAT/05 - Analisi MatematicaSettore MAT/03 - Geometria
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On the additivity of block designs

2016

We show that symmetric block designs $${\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})$$D=(P,B) can be embedded in a suitable commutative group $${\mathfrak {G}}_{\mathcal {D}}$$GD 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 $${\mathrm {PG}}(d,2)$$PG(d,2) and $${\mathrm {AG}}(d,3)$$AG(d,3). In both cases, the blocks can be characterized as the only k-subsets of $$\mathcal {P}$$P whose elements sum to zero. It follows that the group of automorphisms of any such design $$\mathcal {D}$$D is the group of automorphisms of $${\mathfrak {G}}_\mathcal {D}$$GD that leave $$\mathcal {P}$$P in…

Discrete mathematicsAlgebra and Number Theory010102 general mathematics0102 computer and information sciencesAutomorphism01 natural sciencesCombinatorics010201 computation theory & mathematicsAdditive functionDiscrete Mathematics and CombinatoricsSettore MAT/03 - Geometria0101 mathematicsInvariant (mathematics)Symmetric designAbelian groupBlock designs Symmetric block designs Hadamard designs Steiner triple systemsMathematicsJournal of Algebraic Combinatorics
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