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Designs in finite classical polar spaces

Michael Kiermaier, Kai-Uwe Schmidt, Alfred Wassermann

TL;DR

This work extends the theory of $q$-analogs of designs to finite classical polar spaces by defining $t$-$(r,k,\lambda)_Q$ designs within a polar space of type $Q$ and rank $r$, allowing blocks of arbitrary dimension $k$. It develops foundational tools including admissible parameters, derived and residual designs, intersection-number equations, and a Fisher-type inequality, and uses these to classify symmetric designs while proving new structural results. A substantial computational component via the Kramer–Mesner method identifies over $140$ realizable $2$-designs for $q=2,3$ across several polar-space families, including previously unknown parameter sets and $k<r$ cases, with data published openly. Together, these contributions advance both the theory and practical construction of subspace designs in polar geometries, with potential impact on coding theory and finite-geometry applications. $

Abstract

Combinatorial designs have been studied for nearly 200 years. Fifty years ago, Cameron, Delsarte, and Ray-Chaudhury started investigating their q-analogs, also known as subspace designs or designs over finite fields. Designs can be defined analogously in finite classical polar spaces, too. The definition includes the m-regular systems from projective geometry as the special case where the blocks are generators of the polar space. The first nontrivial such designs for t > 1 were found by De Bruyn and Vanhove in 2012, and some more designs appeared recently in the PhD thesis of Lansdown. In this article, we investigate the theory of classical and subspace designs for applicability to designs in polar spaces, explicitly allowing arbitrary block dimensions. In this way, we obtain divisibility conditions on the parameters, derived and residual designs, intersection numbers and an analog of Fisher's inequality. We classify the parameters of symmetric designs. Furthermore, we conduct a computer search to construct designs of strength t=2, resulting in designs for more than 140 previously unknown parameter sets in various classical polar spaces over GF(2) and GF(3).

Designs in finite classical polar spaces

TL;DR

This work extends the theory of -analogs of designs to finite classical polar spaces by defining - designs within a polar space of type and rank , allowing blocks of arbitrary dimension . It develops foundational tools including admissible parameters, derived and residual designs, intersection-number equations, and a Fisher-type inequality, and uses these to classify symmetric designs while proving new structural results. A substantial computational component via the Kramer–Mesner method identifies over realizable -designs for across several polar-space families, including previously unknown parameter sets and cases, with data published openly. Together, these contributions advance both the theory and practical construction of subspace designs in polar geometries, with potential impact on coding theory and finite-geometry applications. $

Abstract

Combinatorial designs have been studied for nearly 200 years. Fifty years ago, Cameron, Delsarte, and Ray-Chaudhury started investigating their q-analogs, also known as subspace designs or designs over finite fields. Designs can be defined analogously in finite classical polar spaces, too. The definition includes the m-regular systems from projective geometry as the special case where the blocks are generators of the polar space. The first nontrivial such designs for t > 1 were found by De Bruyn and Vanhove in 2012, and some more designs appeared recently in the PhD thesis of Lansdown. In this article, we investigate the theory of classical and subspace designs for applicability to designs in polar spaces, explicitly allowing arbitrary block dimensions. In this way, we obtain divisibility conditions on the parameters, derived and residual designs, intersection numbers and an analog of Fisher's inequality. We classify the parameters of symmetric designs. Furthermore, we conduct a computer search to construct designs of strength t=2, resulting in designs for more than 140 previously unknown parameter sets in various classical polar spaces over GF(2) and GF(3).
Paper Structure (9 sections, 18 theorems, 37 equations, 3 tables)

This paper contains 9 sections, 18 theorems, 37 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Designs in polar spaces
  4. Derived and residual designs
  5. Intersection numbers
  6. Fisher's inequality and symmetric designs
  7. Computational construction of designs
  8. Results
  9. Open questions and future work

Key Result

Lemma 1

Theorems & Definitions (33)

  • Lemma 1
  • proof
  • Definition 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 23 more