---
_id: '66731'
abstract:
- lang: eng
  text: "We generalize intersection numbers for combinatorial designs to designs in
    finite meet-semilattices satisfying suitable regularity conditions. While designs
    in regular semilattices go back to Delsarte, our regularity assumptions are weaker
    than his and need not give rise to an association scheme. In this framework, we
    extend Mendelsohn's equations, prove a generalized Singleton bound with Steiner
    systems as equality cases, and determine the block intersection distribution at
    any block of a Steiner system. In particular, this distribution is independent
    of the chosen block.\r\n  Specializing to several classical semilattice families,
    our results recover a number of well-known distributions in coding and design
    theory. In the Hamming and the $q$-Hamming (or bilinear forms) schemes, they give
    the local distance distributions of MDS and MRD codes, respectively. In the Johnson
    and $q$-Johnson (or Graßmann) schemes, they reproduce the block intersection distribution
    of classical and $q$-analog Steiner systems, equivalently the distance distribution
    of diameter-perfect constant-weight codes and diameter-perfect constant-dimension
    subspace codes. For the $q$-Johnson schemes, to the best of our knowledge, this
    result is new. As a further illustration, we apply our theory to designs of perfect
    matchings.\r\n  Our approach provides a unified treatment of these cases in the
    strongest form known in the literature, determining the distribution relative
    to each individual block or codeword, without averaging and without linearity
    or additivity assumptions. Moreover, it identifies the natural double-counting
    objects underlying these distributions, leading to formulas in the regularity
    parameters of the semilattice and avoiding the more cumbersome expressions that
    arise in eigenvalue-based approaches via the ambient association scheme."
author:
- first_name: Michael
  full_name: Kiermaier, Michael
  last_name: Kiermaier
- first_name: Lukas-André Dominik
  full_name: Klawuhn, Lukas-André Dominik
  id: '91965'
  last_name: Klawuhn
  orcid: 0009-0009-7736-4885
citation:
  ama: Kiermaier M, Klawuhn L-AD. Intersection numbers for designs in regular semilattices.
    Published online 2026.
  apa: Kiermaier, M., &#38; Klawuhn, L.-A. D. (2026). <i>Intersection numbers for
    designs in regular semilattices</i>.
  bibtex: '@article{Kiermaier_Klawuhn_2026, title={Intersection numbers for designs
    in regular semilattices}, author={Kiermaier, Michael and Klawuhn, Lukas-André
    Dominik}, year={2026} }'
  chicago: Kiermaier, Michael, and Lukas-André Dominik Klawuhn. “Intersection Numbers
    for Designs in Regular Semilattices,” 2026.
  ieee: M. Kiermaier and L.-A. D. Klawuhn, “Intersection numbers for designs in regular
    semilattices.” 2026.
  mla: Kiermaier, Michael, and Lukas-André Dominik Klawuhn. <i>Intersection Numbers
    for Designs in Regular Semilattices</i>. 2026.
  short: M. Kiermaier, L.-A.D. Klawuhn, (2026).
date_created: 2026-08-17T10:29:36Z
date_updated: 2026-08-17T10:31:03Z
department:
- _id: '100'
external_id:
  arxiv:
  - '2608.14437'
language:
- iso: eng
status: public
title: Intersection numbers for designs in regular semilattices
type: preprint
user_id: '91965'
year: '2026'
...