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Higher incidence matrices and tactical decomposition matrices

Michael Kiermaier, Alfred Wassermann

TL;DR

The paper tackles unifying tactical decompositions with higher incidence matrices to derive Fisher-type inequalities and enable construction of $t$-designs with prescribed automorphisms. It proposes a suite of higher tactical decomposition matrices $R^{(xy)}$, $K^{(xy)}$, $\rho^{(x)}$, $\kappa^{(x)}$ and their averaged forms $W^{(xy)}$, $\omega^{(x)}$, together with transitivity relations and Wilson-type identities. Key contributions include a generalized Fisher inequality for tactical decompositions of both combinatorial and subspace designs, plus an algorithmic framework and a small-case demonstration showing feasibility. The framework extends to $q$-analogs for subspace designs, providing a cohesive toolkit for existence restrictions and automorphism-guided construction of new design families.

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.

Higher incidence matrices and tactical decomposition matrices

TL;DR

The paper tackles unifying tactical decompositions with higher incidence matrices to derive Fisher-type inequalities and enable construction of -designs with prescribed automorphisms. It proposes a suite of higher tactical decomposition matrices , , , and their averaged forms , , together with transitivity relations and Wilson-type identities. Key contributions include a generalized Fisher inequality for tactical decompositions of both combinatorial and subspace designs, plus an algorithmic framework and a small-case demonstration showing feasibility. The framework extends to -analogs for subspace designs, providing a cohesive toolkit for existence restrictions and automorphism-guided construction of new design families.

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.
Paper Structure (11 sections, 15 theorems, 62 equations)

This paper contains 11 sections, 15 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Combinatorial designs
  4. Higher incidence matrices
  5. Tactical decomposition matrices
  6. Higher tactical decomposition matrices
  7. The tactical matrices $R$ and $K$
  8. The tactical decomposition matrices $\rho^{(x)}$ and $\kappa^{(x)}$
  9. The averaged matrices $W^{(xy)}$ and $\omega^{(x)}$
  10. Generalization to subspace designs
  11. Algorithmic use

Key Result

Lemma 3.2

Let $x,y\in\{0,\ldots,v\}$ be integers with $x \leq y$. Then for all $\mathcal{X}\in\mathfrak{P}_x$ and all $\mathcal{Y}\in\mathfrak{P}_y$ we have This can be rewritten as the equality of matrix products

Theorems & Definitions (33)

  • Example 3.1
  • Lemma 3.2
  • Proof 1
  • Lemma 3.3
  • Proof 2
  • Lemma 3.4
  • Proof 3
  • Remark 3.5
  • Lemma 3.6
  • Proof 4
  • ...and 23 more