Glasnik Matematicki, Vol. 58, No. 2 (2023), 181-200. \( \)

HIGHER INCIDENCE MATRICES AND TACTICAL DECOMPOSITION MATRICES

Michael Kiermaier and Alfred Wassermann

Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany
e-mail:michael.kiermaier@uni-bayreuth.de

Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany
e-mail:alfred.wassermann@uni-bayreuth.de

Abstract.   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.

2020 Mathematics Subject Classification.   05B05, 51E05, 52C35

Key words and phrases.   Designs, subspace designs, tactical decomposition

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.58.2.03

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