Search for Combinatorial Objects Using Lattice Algorithms – Revisited

Wassermann, Alfred

doi:10.1007/978-3-030-79987-8_2

Cited by 4 publications

(2 citation statements)

References 46 publications

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“…Solving this system of Diophantine linear equations with the second author's software solvediophant [24] shows that -just as in [18] -exactly one of the eight matrices ρ (1) (namely ρ…”

Section: Algorithmic Usementioning

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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“…Solving this system of Diophantine linear equations with the second author's software solvediophant [24] shows that -just as in [18] -exactly one of the eight matrices ρ (1) (namely ρ…”

Section: Algorithmic Usementioning

confidence: 99%

Higher incidence matrices and tactical decomposition matrices

Kiermaier¹,

Wassermann²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…For the evaluation, we note that κ (0) = (1 1 1 1) and that κ (1) is determined by Lemma 3.2 by the given ρ (1) . The right hand side of the last equation equals  Solving this system of Diophantine linear equations with the second author's software solvediophant [23] shows that -just as in [17] -exactly one of the eight matrices ρ (1) (namely ρ (1) = ρ…”

Section: Algorithmic Usementioning

confidence: 99%

Higher incidence matrices and tactical decomposition matrices

Kiermaier,

Wassermann

2023

Glas. Mat. Ser. III

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In 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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On the classification of unitals on 28 points of low rank

Tonchev

Wassermann

2022

AAECC

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The classification of unitals with parameters 2-(28, 4, 1) according to the 2-rank of their incidence matrices was initiated by McGuire, Tonchev and Ward, who proved that the 2-rank of any unital on 28 points is greater than or equal to 19, and up to isomorphism, there is a unique unital with 2-rank equal to 19. Jaffe and Tonchev investigated the next two 2-ranks, 20 and 21, and showed that there are no unitals on 28 points with 2-rank equal to 20, and there are exactly 4 isomorphism classes of unitals of rank 21. The subject of this paper is the classification of unitals having 2-rank 22, 23 and 24.

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Search for Combinatorial Objects Using Lattice Algorithms – Revisited

Cited by 4 publications

References 46 publications

Higher incidence matrices and tactical decomposition matrices

Higher incidence matrices and tactical decomposition matrices

Higher incidence matrices and tactical decomposition matrices

On the classification of unitals on 28 points of low rank

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