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Large sets of subspace designs

Michael Braun, Michael Kiermaier, Axel Kohnert, Reinhard Laue

TL;DR

This work develops a framework for $q$-analogs of block designs by using decompositions of the Graßmannian into joins and introducing $(N,t)$-partitionable sets to extend large-set recursion to subspace designs. It reports the computer-assisted construction of a $2$-$(6,3,78)_5$ design corresponding to a halving $LS_5[2](2,3,6)$ and proves the existence of two infinite two-parameter series $LS_q[2](2,k,v)$ for $q eq1$ with $v equiv 0,2 mod 4$ and $3\le k\le v-3$ (specifically $v\ge6$, $v ot ot ext{mod }4$ and $k ot ot ext{mod }4$). The approach combines the Kramer–Mesner method with join-based decompositions to generate halvings and large sets, yielding infinitely many nontrivial large sets with $t=2$ and establishing duality and recursive constructions. These results significantly extend the catalog of $q$-analogs of block designs and provide a versatile machinery potentially applicable to network coding and related areas of finite geometry.

Abstract

In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Graßmannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs. We construct a $2$-$(6,3,78)_5$ design by computer, which corresponds to a halving $\operatorname{LS}_5[2](2,3,6)$. The application of the new recursion method to this halving and an already known $\operatorname{LS}_3[2](2,3,6)$ yields two infinite two-parameter series of halvings $\operatorname{LS}_3[2](2,k,v)$ and $\operatorname{LS}_5[2](2,k,v)$ with integers $v\geq 6$, $v\equiv 2\mod 4$ and $3\leq k\leq v-3$, $k\equiv 3\mod 4$. Thus in particular, two new infinite series of nontrivial subspace designs with $t = 2$ are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with $t = 2$.

Large sets of subspace designs

TL;DR

This work develops a framework for -analogs of block designs by using decompositions of the Graßmannian into joins and introducing -partitionable sets to extend large-set recursion to subspace designs. It reports the computer-assisted construction of a - design corresponding to a halving and proves the existence of two infinite two-parameter series for with and (specifically , and ). The approach combines the Kramer–Mesner method with join-based decompositions to generate halvings and large sets, yielding infinitely many nontrivial large sets with and establishing duality and recursive constructions. These results significantly extend the catalog of -analogs of block designs and provide a versatile machinery potentially applicable to network coding and related areas of finite geometry.

Abstract

In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Graßmannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs. We construct a - design by computer, which corresponds to a halving . The application of the new recursion method to this halving and an already known yields two infinite two-parameter series of halvings and with integers , and , . Thus in particular, two new infinite series of nontrivial subspace designs with are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with .

Paper Structure

This paper contains 20 sections, 32 theorems, 50 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. History
  3. Overview
  4. Outline
  5. Dedication
  6. Preliminaries
  7. The subspace lattice
  8. The reduced row echelon form
  9. Subspace designs
  10. Decompositions of $\genfrac{[}{]}{0pt}{}{V}{k}_{q}$ into joins
  11. Joins of subspaces
  12. Paths in the grid graph
  13. Decompositions
  14. $(N,t)$-partitionable sets
  15. Examples of Halvings
  16. ...and 5 more sections

Key Result

Lemma 2.2

Let $i\in\{0,\ldots,v\}$ and $U \leq \mathop{\mathrm{GF}}\nolimits(q)^v$. The canonical matrix of $U$ has a unique block decomposition where $A$ and $C$ are in reduced row echelon form, $A$ has $v-i$ columns and $C$ has $i$ columns. We have

Figures (6)

  • Figure 1: Ordinary join of $K_1$ and $K_2/U$
  • Figure 2: Covering and avoiding join of $K_1$ and $K_2/U_2$
  • Figure 3: Decomposition of $\genfrac{[}{]}{0pt}{}{\mathop{\mathrm{GF}}\nolimits(q)^{10}}{3}_{q}$ into ordinary joins
  • Figure 4: Decomposition of $\genfrac{[}{]}{0pt}{}{\mathop{\mathrm{GF}}\nolimits(q)^{10}}{3}_{q}$ into avoiding joins
  • Figure 5: Decomposition of $\genfrac{[}{]}{0pt}{}{\mathop{\mathrm{GF}}\nolimits(q)^{10}}{3}_{q}$ into covering joins
  • ...and 1 more figures

Theorems & Definitions (77)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 67 more