[2608.14437] Intersection numbers for designs in regular semilattices Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2608.14437 (math) [Submitted on 14 Aug 2026] Title: Intersection numbers for designs in regular semilattices Authors: Michael Kiermaier , Lukas Klawuhn View a PDF of the paper titled Intersection numbers for designs in regular semilattices, by Michael Kiermaier and Lukas Klawuhn View PDF HTML (experimental) 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. 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. 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. Subjects: Combinatorics (math.CO) MSC classes: 05B05, 06A07 (Primary) 05E30, 51E10, 94B65 (Secondary) Cite as: arXiv:2608.14437 [math.CO] (or arXiv:2608.14437v1 [math.CO] for this version) https://doi.org/10.48550/arXiv.2608.14437 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Michael Kiermaier [ view email ] [v1] Fri, 14 Aug 2026 16:21:30 UTC (33 KB) Full-text links: Access Paper: View a PDF of the paper titled Intersection numbers for designs in regular semilattices, by Michael Kiermaier and Lukas Klawuhn View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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