Equations for coefficients of tactical decomposition matrices for $$t$$ -designs

Krčadinac, Vedran; Nakić, Anamari; Pavčević, Mario Osvin

doi:10.1007/s10623-012-9779-y

Cited by 5 publications

(6 citation statements)

References 2 publications

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“…Let where J 4,1 denotes the all-1 -column of length 4 . In fact, it is clear that this mosaic allows a "chained" tactical decomposition (see [2]) of all copies of affine planes involved in this construction.…”

Section: Further Constructions Of Mosaicsmentioning

confidence: 99%

Mosaics of combinatorial designs

Gnilke

Greferath

Pavčević

2017

Des. Codes Cryptogr.

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Abstract. Looking at incidence matrices of t -(v, k, λ) designs as v×b matrices with 2 possible entries, each of which indicates incidences of a t -design, we introduce the notion of a c -mosaic of designs, having the same number of points and blocks, as a matrix with c different entries, such that each entry defines incidences of a design. In fact, a v×b matrix is decomposed in c incidence matrices of designs, each denoted by a different colour, hence this decomposition might be seen as a tiling of a matrix with incidence matrices of designs as well. These mosaics have applications in experiment design when considering a simultaneous run of several different experiments. We have constructed infinite series of examples of mosaics and state some probably non-trivial open problems. Furthermore we extend our definition to the case of q -analogues of designs in a meaningful way.

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Section: Further Constructions Of Mosaicsmentioning

confidence: 99%

Mosaics of combinatorial designs

Gnilke

Greferath

Pavčević

2017

Des. Codes Cryptogr.

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“…Equations for coefficients of tactical decomposition matrices for block designs are well known [9] and they were used for constructions of many examples of block designs (listed in [13]). These equations were generalized for any t ≥ 1 in [12]. In this article we introduce tactical decompositions of designs over finite fields for t = 2.…”

Section: Tactical Decompositions Of Designs Over Finite Fieldsmentioning

confidence: 99%

Tactical decompositions of designs over finite fields

Nakić

Pavčević

2014

Des. Codes Cryptogr.

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Abstract. An automorphism group of an incidence structure I induces a tactical decomposition on I. It is well known that tactical decompositions of t-designs satisfy certain necessary conditions which can be expressed as equations in terms of the coefficients of tactical decomposition matrices. In this article we present results obtained for tactical decompositions of q-analogs of t-designs, more precisely, of 2-(v, k, λ 2 ; q) designs. We show that coefficients of tactical decomposition matrices of a design over finite field satisfy an equation system analog to the one known for block designs. Furthermore, taking into consideration specific properties of designs over the binary field F 2 , we obtain an additional system of inequations for these coefficients in that case.

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“…While our approach is presented for classical block designs, it can also be adapted to subspace designs in a straightforward way. The interplay to the equations in [19] deserves further investigation.…”

Section: Introductionmentioning

confidence: 99%

“…The majority of publications on the construction of designs with tactical decompositions are based on the point-block incidence matrix and only involve constraints derived from the property of a 2-design. The only exceptions we are aware of are the articles [18,19,20], where constraints are given for the tactical decomposition of the point-block incidence matrix of a t-design of general strength t ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

Higher incidence matrices and tactical decomposition matrices

Kiermaier¹,

Wassermann²

2023

Preprint

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In memory of Professor Zvonimir JankoIn 1985, Janko and Tran Van Trung published an algorithm for constructing symmetric designs with prescribed automorphisms. This algorithm is based on the equations by Dembowski (1958) for tactical decompositions of point-block incidence matrices. In the sequel, the algorithm has been generalized and improved in many articles.In parallel, higher incidence matrices have been introduced by Wilson in 1982. They have proven useful for obtaining several restrictions on the existence of designs. For example, a short proof of the generalized Fisher's inequality makes use of these incidence matrices.In this paper, we introduce a unified approach to tactical decompositions and incidence matrices. It works for both combinatorial and subspace designs alike. As a result, we obtain a generalized Fisher's inequality for tactical decompositions of combinatorial and subspace designs. Moreover, our approach is explored for the construction of combinatorial and subspace designs of arbitrary strength.

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Equations for coefficients of tactical decomposition matrices for $$t$$ -designs

Cited by 5 publications

References 2 publications

Mosaics of combinatorial designs

Mosaics of combinatorial designs

Tactical decompositions of designs over finite fields

Higher incidence matrices and tactical decomposition matrices

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