---
res:
  bibo_abstract:
  - "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.@eng"
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Michael
      foaf_name: Kiermaier, Michael
      foaf_surname: Kiermaier
  - foaf_Person:
      foaf_givenName: Lukas-André Dominik
      foaf_name: Klawuhn, Lukas-André Dominik
      foaf_surname: Klawuhn
      foaf_workInfoHomepage: http://www.librecat.org/personId=91965
    orcid: 0009-0009-7736-4885
  dct_date: 2026^xs_gYear
  dct_language: eng
  dct_title: Intersection numbers for designs in regular semilattices@
...