Intersection numbers for designs in regular semilattices

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.

• The paper develops a double-counting framework for intersection numbers and generalized Mendelsohn equations in regular semilattices without requiring association schemes, eigenvalue methods, or linearity.
• It proves a generalized Singleton bound, with equality precisely for Steiner systems, and derives a block intersection distribution determined only by the design strength and semilattice parameters.
• The framework extends to Johnson, Hamming, q-analog, and perfect-matching semilattices, yielding new results for q-Steiner systems and unrestricted MRD codes while exposing open existence problems.

Overview

The paper develops a general theory of intersection numbers for designs in finite regular meet-semilattices, unifying and extending several classical results in combinatorial design and coding theory. The framework originates with Delsarte's regular semilattices (2608.14437), but the authors deliberately weaken Delsarte's regularity axioms: their posets need not satisfy the property (R$\pi$) that forces the top level to carry an association scheme. This relaxation is not cosmetic — the semilattice of perfect matchings of a complete graph, treated in the final section, is regular in the authors' sense but not Delsarte-regular, so the theory genuinely extends beyond the association-scheme setting.

The main contributions are threefold: a generalized system of Mendelsohn equations together with a Köhler-type parametrization; an extended Singleton bound whose equality cases are precisely Steiner systems; and an explicit formula for the block intersection distribution at any block of a Steiner system, showing in particular that this distribution is independent of the chosen block. All three results are proved by direct double counting in the regularity parameters of the semilattice, without eigenvalue computations and without linearity, additivity, or averaging assumptions.

Regular posets and the weakened axioms

The authors work with a finite poset $(X,{\leq})$ with bottom element $\bot$, partitioned into levels $X_0,\ldots,X_n$ by the height function, where the top level $\Omega = X_n$ plays a distinguished role. Regularity is expressed by three counting conditions: (R$\theta$) requires the number of top-level elements above a fixed element of $X_r$ to depend only on $r$; (R$\mu$) requires the number of elements of $X_s$ between an element of $(X,{\leq})$0 and a top-level element to depend only on $(X,{\leq})$1; and (R$(X,{\leq})$2) requires the number of elements of $(X,{\leq})$3 below a fixed element of $(X,{\leq})$4 to depend only on $(X,{\leq})$5. The resulting parameters $(X,{\leq})$6, $(X,{\leq})$7, and $(X,{\leq})$8 govern all subsequent formulas.

Compared with Delsarte's setting, the authors replace his property (R$(X,{\leq})$9) with the weaker (R$\bot$0), and most statements survive even without (R$\bot$1). Two structural facts are established early: a finite regular poset need not be graded (a small counterexample is exhibited), and the role of $\bot$2 in (R$\bot$3) cannot be replaced by an arbitrary intermediate level, even for graded regular posets — a second explicit counterexample is given. The $\bot$4-matrix is upper triangular with ones on the diagonal and determinant one, so its inverse $\bot$5 is integral, upper triangular, and provides a $\bot$6-inversion principle used throughout.

Delsarte's stronger axioms are reviewed for context: for a Delsarte-regular $\bot$7-semilattice, the relations on $\bot$8 defined by the height of the meet form a symmetric association scheme. The authors stress, however, that their results never invoke this structure.

Designs in regular posets

A design is modeled as a complex-valued weight function $\bot$9, with subsets recovered via characteristic functions. The function $X_0,\ldots,X_n$0 is a $X_0,\ldots,X_n$1-design of index $X_0,\ldots,X_n$2 if $X_0,\ldots,X_n$3 is constant over $X_0,\ldots,X_n$4. A double-counting argument shows that a $X_0,\ldots,X_n$5-design is automatically an $X_0,\ldots,X_n$6-design for all $X_0,\ldots,X_n$7, with index $X_0,\ldots,X_n$8 — a fact Delsarte originally derived algebraically from the dual distribution, requiring his stronger hypotheses. For integer-valued designs, the authors obtain divisibility conditions in three equivalent forms, the cleanest being $X_0,\ldots,X_n$9. In the four classical semilattice families, these specialize to the standard integrality conditions for block designs, subspace designs, and orthogonal arrays; notably, for the Hamming and $\Omega = X_n$0-Hamming semilattices the conditions are trivial.

Generalized Mendelsohn equations

Adding a meet-semilattice structure, the intersection numbers $\Omega = X_n$1 of $\Omega = X_n$2 with respect to $\Omega = X_n$3 count top-level elements $\Omega = X_n$4 with $\Omega = X_n$5, weighted by $\Omega = X_n$6. The central result is the generalized Mendelsohn system: for a $\Omega = X_n$7-design and any $\Omega = X_n$8,

$\Omega = X_n$9

Via $\theta$0-inversion this is equivalent to the Köhler parametrization, which expresses the low intersection numbers $\theta$1 for $\theta$2 in terms of the high ones $\theta$3 — a form that is especially effective when few high numbers are nonzero, as is the case for Steiner systems. Specializing to the Hamming and $\theta$4-Hamming semilattices, the authors state the resulting system as a corollary and note that, for the $\theta$5-Hamming case (and the Köhler form for the Hamming case), no published version appears to exist. For the Johnson family the result recovers the classical equations of Mendelsohn, Goethals, Oberschelp, Köhler, Harnau, and, in the $\theta$6-Johnson case, Kiermaier and Pavčević; for orthogonal arrays it reproduces the Bose–Bush system long used in non-existence arguments, such as the non-existence of the $\theta$7-$\theta$8 design.

Singleton bound and the Steiner intersection distribution

Codes are defined via the packing condition $\theta$9 on $X_r$0, which the authors show coincides with the metric condition $X_r$1 for the meet-height distance. An instructive counterexample built from the Petersen graph shows that $X_r$2 need not satisfy the triangle inequality even for short graded regular $X_r$3-semilattices — so the "distance" is genuinely weaker than a metric, yet the coding-theoretic machinery still functions.

The generalized Singleton bound reads $X_r$4, with equality if and only if $X_r$5 is a Steiner system of strength $X_r$6. The proof is a one-paragraph double count. The main theorem of this section then gives, for a Steiner system $X_r$7 of strength $X_r$8 and any block $X_r$9:

$r$0

Two features deserve emphasis. First, the distribution is determined by $r$1 and the regularity parameters alone, independent of the particular Steiner system and of the chosen block. Second, the result holds for arbitrary (unrestricted) codes: no linearity, no duality, no association scheme. In coding language it determines the local distance distribution $r$2 at each individual codeword of a code attaining the Singleton bound — strictly stronger than Delsarte's inner distribution, which is only the average.

Specializations

Specializing to the Johnson and $r$3-Johnson semilattices yields an explicit binomial/Gaussian-coefficient formula for the block intersection distribution of Steiner and $r$4-Steiner systems. For ordinary Steiner systems this recovers known results (Mendelsohn, Goethals); for subspace designs the authors state that both the uniqueness statement and the formula are new. As illustrations, the intersection distribution of a $r$5-$r$6 design is computed as $r$7, and that of a $r$8-$r$9 binary $\mu$0-Steiner system as $\mu$1.

For the Hamming semilattice, Steiner systems are exactly MDS codes, and the theorem recovers the local distance distribution of MDS codes over arbitrary alphabets — a result previously assembled only from partial sources (linear codes via Assmus–Mattson–Turyn, Kasami–Lin–Peterson, and Forney; unrestricted codes via Goethals and Heise–Quattrocchi). For the $\mu$2-Hamming semilattice, Steiner systems are MRD codes, which exist for all admissible parameters via the Gabidulin construction. The theorem recovers the local distance distribution of unrestricted MRD codes; previously this was available only by combining Delsarte's eigenvalue-based inner distribution for linear codes with the distance-homogeneity established by Ravagnani for the unrestricted case. The double-counting route presented here is, to the authors' knowledge, the first direct combinatorial proof of the formula in that generality, and it avoids the generalized Krawtchouk polynomial expressions inherent to the association-scheme approach.

Designs of perfect matchings

As a demonstration that the weakened axioms matter, the authors analyze the semilattice of matchings of $\mu$3 under intersection, with perfect matchings as the top level. This semilattice is graded and regular, with parameters involving double factorials, but it is not short (a matching with $\mu$4 edges has a unique perfect completion), and for $\mu$5 it is not Delsarte-regular: the authors exhibit explicit matchings showing $\mu$6 is not well-defined. Designs of strength $\mu$7 here are precisely $\mu$8-factorizations; Steiner systems of strength $\mu$9 exist via Cameron's hyperoval construction, but no nontrivial $X_s$0-designs with $X_s$1 are known, and the existence problem is open. The Mendelsohn equations and the Steiner intersection distribution specialize to new explicit formulas. A candid limitation is noted: unlike in the classical cases, evaluating the intersection formula for small parameters always yields non-negative values, so it does not yield non-existence results — in particular, the known non-existence of a $X_s$2-Steiner system in $X_s$3 does not follow from the intersection numbers.

Limitations and open questions

Several boundaries of the framework are stated plainly. The regular-poset axioms do not capture all design-like structures in the literature — designs in finite classical polar spaces and certain matching designs fall outside it. The map $X_s$4 is generally not a metric, so the coding-theoretic interpretation relies on the packing definition rather than metric properties. The divisibility conditions are trivial in the hypercubic families, limiting their diagnostic value there. For perfect matching designs, the intersection numbers provide no non-existence obstruction, the existence problem for $X_s$5 is entirely open, and no construction beyond Cameron's $X_s$6-designs is known. More broadly, concrete Steiner systems remain scarce: in Johnson semilattices only finitely many explicit constructions of strength $X_s$7 or $X_s$8 are known (the Witt designs among them), none with $X_s$9; in $(X,{\leq})$00-Johnson semilattices the only known parameter set with $(X,{\leq})$01 is $(X,{\leq})$02-$(X,{\leq})$03. The paper leaves open whether the double-counting approach can be pushed to structures such as polar-space designs, and whether the matching-semilattice intersection numbers can be supplemented by additional constraints to obtain non-existence results.

Conclusion

The paper provides a unified, elementary treatment of intersection numbers, Mendelsohn equations, the Singleton bound, and Steiner intersection distributions in a class of regular semilattices strictly broader than Delsarte's, requiring no association scheme, no duality, and no linearity. Its formulas are expressed directly in the regularity parameters, and in the $(X,{\leq})$04-Johnson and $(X,{\leq})$05-Hamming settings several of the resulting statements — the subspace Steiner intersection distribution and the unrestricted MRD local distance distribution via direct counting — appear for the first time in the literature. The perfect matching application confirms that the added generality is substantive rather than formal.